So I've got two possible theories about the dog villager Lucky. One's very simple, that he and Walker are brothers. The other's more...crazy.
Before I explain it, I have to give thanks to this great video by Daiton. I highly recommend watching this if you're interested in Lucky. It's what got me to think more in depth about this. Nookipedia's was also helpful in research. I'll be referring to their articles on Lucky, Walker, Gulliver and Ankha for this.
I'll explain my theory/head canon under the cut. Warning, it gets pretty crazy.
I'll start with a rather well known theory involving Lucky. It's the theory that he's actually the undead version of Walker, another lazy dog villager. People speculate this due to similarities and connections between the two.
Also of note is that they share the same fear (Mummy mask) and coffee order (Mocha, The regular amount, Two spoonfuls). This has led some people to think they are the same dog. However, others aren't so convinced. It's easy to be skeptical because:
Their houses are completely different
They have different birthdays (Lucky's is November 4, Walker's is June 10)
Walker's eyes are blue while Lucky's visible eye is yellow
Most importantly, both can be and have been seen on the same village/island
My take on this theory intends to explain these.
For starters, what do we canonically know about Lucky? What is there that we can confirm? Surprisingly enough, his e-reader card flat out reveals why Lucky is bandaged.
Lucky took luck into his own hands when he went on that fateful fishing trip last week. Between casting his rod too far, snagging his line on a rock, and losing his balance and falling overboard, Lucky's luck held true to form.
This is expanded on in The Crossing Guardian, an official website for Animal Crossing back in the 2000s.
Of note here is that he was fishing, which is Walker's hobby in Wild World. Also of note is that apparently Lucky and Gulliver are (or at least were) friends. Gulliver is another character in the Animal Crossing series, a special NPC who has appeared in all the games. He has referenced places in other Nintendo games like the Mario series and The Legend of Zelda.
Now you might think that's just a one off gag, but in Wild World and City Folk (and the anime film based off of Wild World), Gulliver is actually depicted as an astronaut with a UFO that you can shoot down.
Keeping in mind how Gulliver has mentioned his crewmates before ("I'm sure my crewmates are looking for me"), my theory is that Gulliver and Lucky (or more accurately Walker as he would've been known as at the time) were crewmates on a ship that travels through space and time. Due to traveling through space and time, Gulliver experiences the games in a slightly different order than we do: Wild World and City Folk are first since he's an astronaut flying around in those. His crewmates get sick of him and throw him overboard, leading to New Horizons where he's lost and trying to get his communicator parts. Gulliver tries to reach his crew and fails, eventually leading to Population Growing and New Leaf. This is why in all the games Gulliver washes up on the shore (I swear this ties back to Lucky).
When Walker got upset with the other crewmates over the treatment of Gulliver, he too was thrown overboard. He would eventually fish to take his mind off of being betrayed and end up suffering the accident. He would be bandaged up and taken to the hospital. This is where Ankha comes in.
Now with Ankha it's been rumored that she's been around since Cleopatra, as her e-reader card and Pocket Camp description state.
Ankha's face shines with beauty and the sparkle of the youth, but she claims she was a cat in Cleopatra's court. That would make her 3,000 years old! How many years do nine lives make?
Ankha is a cat veiled in mystery... She may look young, but some have said she might actually be 3,000 years old! If that's true, she's holding up really well.
If that's the case it's entirely possible she has magic. And she's also known to love mummies thanks to Happy Home designer.
I want to live surrounded by ancient artifacts because I really adore mummies!
I think, on November 4, Ankha came across Walker and healed or revived him with her magic. When Walker "miraculously" recovered he became known as Lucky due to how lucky he is. Walker came to truly believe he was Lucky, which is why he believes his birthday is on November 4. This is also why Lucky's house has an Egyptian theme in most of the games and why his skill in New Leaf is "forgetting stuff". The supernatural nature of the whole thing is also why Lucky's visible eye is yellow.
This part gets REALLY crazy: Another version of Walker would also get thrown overboard by the crewmates, on a very specific date... December 31, 2030, the furthest Population Growing goes. Because there were two of the same person (or dog) in the same world, a paradox was created. In the timeline Population Growing would be overtaken with Doubutsu no Mori e+.
This is why Doubutsu no Mori e+ is where Walker debuts and why villagers first introduced in that game appear in later games. This is also why Walker and Lucky can appear in the same town.
So the timeline now goes: Doubutsu no Mori e+, Animal Crossing Wild World, Animal Crossing City Folk, Animal Crossing New Leaf, Animal Crossing New Horizons
TLDR: Walker and Gulliver were crewmates in a spaceship that travelled across space and time. Gulliver would get mistreated by the crewmates. When Walker got upset over Gulliver's treatment he got thrown overboard. He would eventually get into a fishing accident and get hospitalized. On November 4 he would get turned into a mummy by Ankha, and turn into Lucky. Lucky would forget he ever was Walker. Another Walker is thrown into the same world as Lucky, causing a paradox and a shift in the timeline to accommodate.
