Right, so. Akai and Rei, and what makes them so fascinating to me.
First of all, go check out this post by @the-red-thread-that-strangles about the whole lone wolf stereotype that Akai seems to be written as (and how it doesn't fit him at all, but actually fits Rei instead). The Rei section isn't a connection I had made and it really just fuels my thoughts even further.
Onto my thoughts. Let's start with the similarities between them.
You have two men that are undercover cops working (or who previously worked, in Akai's case) inside the Organization.
Their motivations for doing so, at least to start, are the deaths of a loved one: Tsutomu for Akai, and Elena for Rei (Akai's father, and something of a mother figure for Rei). (And the fact that Elena is actually Akai's aunt is another fascinating detail that further entwines Akai and Rei's stories.)
They both experience the death of another loved one at the Organization's hands in the more-present day that leads them on even more of a revenge quest: Akemi, killed by Gin, and Hiro, who killed himself three years ago to protect his loved ones, Rei included. Both Akemi and Hiro are traitors to the Organization (at least, that's how the Organization sees them; the argument can be made that neither of them are traitors because neither of them wanted to be loyal to the Organization in the first place).
They are both skilled detectives, fighters (with differing specialties), and marksmen (though we don't know how Rei does with long-range). Both are mixed race (half or more Japanese).
And they both have canine-themed "nicknames" for each other. Rei refers to Akai as an FBI dog in the Mystery Train arc, and Akai refers to Rei as a wolf on two occasions: towards the end of Scarlet Return and at the very end of file 1150 (the second most recent case as of writing this post). (Part of this goes back to Iris' Lone Wolf post, but there's more depth there that I'll bring up in a bit, because how they refer to each other is notable.)
(They both tried to save Hiro the night he died, and they both failed. And it's a failure that eats at both of them. I'm not sure that entirely fits the point of this post, but it's worth noting.)
So unless I've forgotten something (which is entirely possible, as I didn't make an outline and I'm mostly writing this off the top of my head), that's it for similarities.
As for the differences, there are many. To keep things as simple as possible, I'll try to follow the same order as much as I can.
Family:
Akai obviously has many family members brought up throughout the manga: his father, Tsutomu, killed by the Organization when he was young. Elena, his aunt, and her husband, Atsushi, also killed around the same time as Tsutomu. Akemi, killed by Gin in basically present-day.
But he also has several living family members: Shiho (who he doesn't know is his cousin), his mother, Shuukichi and Masumi, all of whom are relatively important to the story.
On the other side of things, you have Rei, who has never had any family mentioned. The closest argument you can make is 1) Elena as a pseudo-mother (which is how I see their relationship, because I don't buy into him being "in love" with her, when he's like 11-12 years old), Hiro, his best friend/possible brother figure, or the other WPS who could be see as brothers as well. None of those are biological family, and even within the story, none of them are referred to as family.
And, notably, they're all dead. Rei does not have any living family members (or family-adjacent people) that have ever been brought up in the story. The most we know is that he's half-Japanese. Presumably, his father is Japanese (due to his family name being Furuya) and his mother is foreign. That's all we've got.
(Obviously most of this is, again, brought up in Iris' post. I felt it was worth reiterating here, as I'm using it for contrast.)
Work:
Within the Organization, Akai, then known as Rye, was a sniper (we're told he's the best, at least in M26; this may have also been brought up in the manga). I don't believe we're ever told if he doesn't anything besides the sniper gig, but I may have forgotten. Either way, his primary function is as a killer.
His identity got exposed two years prior to canon, so he left, and only recently returned to Japan. While he doesn't often work in an FBI capacity directly (due to having faked his death), when he does, there's usually some sniping involved.
As for his role within the FBI, he doesn't seem to have any sort of elevated position. James is his supervisor (even if he mostly does what he wants, rather than following orders), and he works with the team of James, Jodie and Camel (and occasionally other unimportant FBI agents, when it's specifically relevant to a case, such as the FBI serial murders). He, as far as we're aware, is not anyone's boss.
And then there's Rei, aka Bourbon, who works in reconnaissance. Gin, Kir, and other Organization agents liken him to Holmes with how effective of a detective he is, to the point that they seem to be a bit scared of him. He's the one going around disguised as Akai (by his own choice, but with Vermouth help, so she must have given some sort of approval) to investigate whether he's really dead. He's the one with a long-term undercover assignment as Amuro to first look for Sherry, and then look into Shinichi.
So Bourbon has more of a "brainy" role versus Rye's "brawn," if you want to look at it as like a spectrum.
As for Furuya Rei, he has at least one subordinate: Kazami. He has other men that have worked with him, but we don't know their specific roles within the PSB. He does take orders from Kuroda, but we don't really know Kuroda's place within the PSB, either (he's superintendent over the MPD, but, structurally, I'm not sure that also puts him over the NPA-PSB as well). Rei has an actual leadership role (and not just because people follow him), and is usually the one giving orders when Kuroda's not involved.
Appearance:
Akai is a black-haired, pale-skinned man whose usual outfit (as himself) is a black leather jacket, a dark navy shirt, and brown(?) pants. He rarely deviates from this when not in disguise as Okiya.
Rei is a blond-haired, dark-skinned man whose usual outfit (for work) in a grey suit, white shirt, and teal tie. Sometimes the tie color changes, and only once has the rest of the suit changed when he's working in PSB capacity. One a couple of occasions, when he's home alone, he's been seen wearing a white t-shirt.
So they do even have the whole contrasting dark/light motif going on. (Although whether they actually fit what you normally expect from that is another story entirely. Hint: they don't.
And argument could be made for yin and yang, however, which I feel is much more fitting for them. Opposite but complementary, two halves of a whole.
...although that may just be getting into soulmatism. Which isn't really what this is about, but hey. If the shoe fits.)
Dogs:
Rei calls Akai 'FBI dog'. Akai calls Rei 'wolf'.
This is actually very interesting to me because it's sort of the opposite 'nickname' of what you'd expect looking at them surface-level.
Dogs are domesticated. They're loyal, can be trained to follow rules, generally friendly and outgoing, and are generally beloved to the point where we have a whole 'dog people' thing.
Wolves are wild animals. They are untrained, undomesticated, and, due to being, you know, wild animals, are a bit more unpredictable. There's a whole genre of horror that focuses on people who turn into dangerous wolves (werewolves).
So, based on that, you'd look surface-level and think that the one who Rei gets angry about working outside of his jurisdiction (doesn't follow rules, 'undomesticated'), who stalks his prey on account of being a sniper, who the Organization isn't necessarily afraid of, but certainly they're wary -- you'd think that person would be the wolf.
And you'd look at the ideas we're present with Rei and think 'definitely dog.' He doesn't like that Akai breaks the rules, so he must follow them pretty rigidly. He seems to be very well-liked, and he acts very friendly a lot of the time. And he's very loyal to his master (his country, which he loves so much).
But the thing is that that's just surface-level, and doesn't actually describe them all that well. They have it right with what they call each other (because they actually know each other far too well, definitely better than most other people know them).
Akai may break the rules he doesn't care about, but he has his own set of internal rules that he follows, and those generally seem to be mostly about protecting the ones he cares for, which is a circle that has grown fairly huge within the series these days (can refer back to Iris' post here again). Nearly everybody he comes into contact with loves him. Yukiko even calls him Shuu-chan, for crying out loud, and he doesn't seem to object to it. And Vermouth calls him a Silver Bullet - the thing that kills (were)wolves. He is the very essence of a dog when you get past the "Lone Wolf" stereotype he got incorrectly saddled with.
And Rei? He may complain about Akai's rulebreaking, but he's even worse. He's the one stealing keys and breaking into a civilian home to confront Akai. His Amuro persona is beloved (and I guess Kazami might care a little about him), but that's just a mask he wears to get people to like (and more importantly, trust) him. He's the one more liable to snap and bite someone (metaphorically; he's not likely to literally bite someone in canon). And speaking of stalking prey - that's exactly what he's been doing to Akai for months.
So Akai is, in fact, the dog, and Rei's the wolf, despite what they seem to present to other people.
Personality:
So, like I said before, light and dark don't really fit them because rather than being dark/light themed in like personality or personal principles, they're both pretty grey. But you know what does fit them? Fire and ice. (And it's complicated. We'll get to that.)
Akai's the ice, with his Cool Guy personality. He keeps his emotions locked up tight, and rarely shows anything deeper than completely mild. He doesn't even smile that often (so it's notable when he does), and doesn't seem to like to talk about the important things, only sharing bits of himself when it's necessary, even to those closest to him. He is patient on account of being a sniper.
And Rei is all fire. He's prone to violent, angry outbursts (especially when it comes to Akai). He's impatient (though a lot of that falls back to Akai being a problem for him; he's a lot less chaotic when Akai isn't involved). In the past, we're shown that he warms to people pretty quickly (he made friends with Hiro during a time when Hiro wouldn't even talk to him yet, and he and Matsuda beat the hell out of each other and then shared their tragic backstories and became besties. It didn't take much for him to warm to Elena, Date and Hagi, either).
Now, here's where it gets complicated. In present day, Rei doesn't warm to people easily. The most notable connection he's made is with Conan, which didn't start off as him being fond of the kid, but rather curious about him. The fondness came later. (And it's very one-sided, but that's beside the point.) He just doesn't make those connections anymore, partially due to being undercover, but also probably due to having lost everyone he loves. So, in that respect, it seems like he's someone warm who became cold.
And then Akai, someone who is generally cold, but has his moments where it seems like he's warming up. He took to Conan quickly, and aside from him, Rei is one of the few people that gets him to smile (if we set aside the Okiya persona). A cold person, who becomes warmer around the right people.
Soulmatism. Complementary opposites.
I think that's everything I wanted to bring up, but feel free to bring up other stuff. I just think they're neat.