It's actually a good thing that the Buddy incarnation was just that, an incarnation. Otherwise, he would've been hung up on those owners, even if all they ever did was neglect him. You see, in real life, dogs who are abandoned become nervous and miss their original owners. They'll either stay where they are, or go on a manhunt for their family even if it's futile. But Bailey already knows what it means to have a worthwhile family who loves and plays with him, and that wasn't it. So instead of fruitlessly chasing a family that doesn't want him upon abandonment, Bailey finds it easier to cut ties with them and find a true forever family.
Each of Bailey's lives corresponds to what he found dissatisfying about his previous one, or wished more out of it. Each of those lives has a purpose in store for him.
The stray life is one where he had fun, but wasn't happy with how short or meaningless it was, setting off his first ever reincarnation.
His reincarnation as Bailey is where he had a long life and a meaningful purpose (to make Ethan happy). Only, he found he was dissatisfied to find his purpose was somewhat unfulfilled (Ethan was unhappy at his passing).
His reincarnation as Ellie was where he could fulfill his purpose (to protect and rescue people) and got to utilize his rescue skills from when he saved Ethan from a fire. The part he wasn't happy about was that that life "wasn't fun".
Thus, his reincarnation as Tino gives him a more fun life, where his purpose is to protect and love Roxy. Its one hitch is her untimely passing, leaving him lonely. But it prepares Bailey to empathize how Ethan needs Hannah in his life.
The final reincarnation we see (as Buddy) is one where Bailey fulfills his first purpose (making Ethan happy) by bringing Hannah back into his life.
It might seem odd that "Horse Dog" seemed to be the same one Bailey remembered from his earlier life despite it being several reincarnations later. But donkeys can actually live a surprisingly long time, over 40 years. And it was a young donkey when Bailey met it. So while it would be an old donkey near the end of it's life it's actually quite possible that several incarnations and decades later that donkey would still be alive and at the farm.
Fridge Horror
Bailey's first life is as a stray. He's scooped up and put into an animal control van. The scene fades out, then fades back into his next life as Bailey. Younger viewers might easily fail to realize that that the stray was likely taken to a shelter and eventually euthanized.
Bailey is shot and killed in his incarnation as Ellie, the police dog. Recall that throughout this segment, he has sensed that his handler is sad and lonely because he's missing someone. Indeed, we see a picture of the man and a woman, presumably his wife, on his nightstand. While it's not clear if the woman died or divorced him, either way, this poor man just suffered TWO major losses in his life.
Bailey spends the first few years of his Buddy incarnation being neglected and eventually abandoned by his owners. It's implied that the guy abuses his girlfriend/wife and that's the very reason why she can't care for him properly. Then, the jerk lies to the girl that he'll make sure to give him to a good home. So aside from wondering what terrible fate was in store for that girl for the rest of her life, the poor thing might be consoling herself with the completely false belief that at least the dog was okay.
While it was in no way thanks to her awful boyfriend, at least Buddy was eventually okay, considering he eventually found Ethan and Hannah and lived a good rest of his life with them.
