Underneath the "Read More" I'm gonna go into detail as to why he's befriended each one of them, though I don't have what he would've named (or what they would call themselves) figured out, so if you wanna suggest ideas, go right ahead!
Now onto the show.
Jason Todd's favorite gift he's ever received was from his mom before everything really went to shit. It was his birthday, his father spending it out on the streets, and Catherine came back home that day with a Lillipup- they're common pokemon, and some were recently hatched and up for adoption, and taking care of one compared to say a Zubat is relatively easy. Obviously, he comes to adore her.
Not all good things last, especially not in Jason's case, and eventually his father's arrested and his mom spends more and more time spent between getting money and doing drugs, and while he loves his mom he can't save her, and neither can Lillipup, though her habit of picking up food on the streets and sharing it with Jason does help the two of them when Catherine is particularly unreachable. Then she dies, and the two end up on the streets.
Shortly after sticking it in the streets, Lillipup evolves, becoming Herdier, refusing to go into her pokeball to attempt to keep Jason safe. Jason learns to trust her instincts, but the two still get caught by Batman when they're stealing his tires.
When Jason becomes Robin, she's right there alongside him, refusing to listen to Bruce and finding every way through the security. Eventually Batman relents and allows her on the patrols, partially because the likelihood of people connecting the two Herdiers together, especially when Herdiers are one of the most common police pokemon besides Growlithe, is slim to none.
When Jason goes to board a plane to Ethiopia, he's required to put Herdier into her pokeball. He's too worried about his birth mother dealing with the Joker to let her out. By the time Bruce releases her, it's too late. He's already dead.
She refuses to go back into her pokeball, and on the day of his funeral is sure to break it while fighting over it. Herdier spends the next few months refusing to leave the cemetery, keeping watch over his grave. She knows he’s dead, knows that Jason’s people are worried about her, but she refuses to leave his side.
When he starts digging, she’s right there besides him, throughout his time in the hospital labeled as a John Doe, refusing to let Talia take him without her, the first face he sees after waking from the Lazarus Pits. While he trains, she evolves one final time into a Stoutland, and while she might be a more common pokemon than her counterparts, she’s fiercely protective of Jason and the kids of Crime Alley to the point that everyone knows better than to mess with her.
Unfortunately, my concepts for the other two aren’t as emotional.
Instead of getting the All-Blades, Ducra allows him to befriend Doublade during his time meditating with her, and eventually the pokemon decides that Jason is a worthy trainer that it shall follow. Considering the point of Jason going to Ducra, I’d say Doublade is stubborn as all get out, refusing to back down from a fight unless it would result in an injury to a friend, but secretively kind to those it and Jason care about, most notably stealing food for Stoutland.
Pangoro was a leader of wild pokemon living in East Uptown Gotham- when Red Hood comes along, it becomes obvious that they either have to get along or duke it out. The two of them start of more as backup for the other leader of a partner gang, but over time Pangoro decides that Red Hood is worth his weight, and while refusing to be captured as a “proper pokemon”, basically lives alongside Doublade and Stoutland, going and checking on his crew when needs be.
Yeah, the last two aren’t nearly as long as Stoutland, but in my defense, she’s the one that has lived her entire life alongside Jason, so it makes sense that she’d have so much more written.