Look, the BB Bandits are basically just Team Rocket. They serve the exact same purpose in the story, have the same general membership, and no-one ever gets quite enough characterization to be able to say "these are reasonably distinct characters." However! In one of the only pieces of background information about these three, we learn that Rex is not a vivosaur, nor a Dinurian. The BB Bandits simply do not know where he is from. Somehow, they can understand what he says, but I guess they never bothered to ask him that. Certainly, during the brief period we can understand him, he never speaks on the subject. And, despite this being a subtractive effort compared to, say, Meowth's whole deal, I kind of love it? Like this game has literal aliens and revived dinosaurs, the whole game is basically about non-human lifeforms, yet at no point is Rex ever even passingly explained. There's just some talking dog who speaks in a posh British accent, just deal with it. I'm too lazy to look up an article about how the whole "lore" thing, i.e. explaining every single bit of backstory, kind of sucks, but Rex's backstory is absolutely something that would be put into an audio log in a different sort of game. But it's not, and now people can just speculate about this entirely out of place character forever, no interpretation demonstrably true or false.
And given what kind of website this is, I'd like to offer my own hypothesis. Consider that the crashed spaceship on the Secret Island was there since at least the Jurassic period. Therefore, mayhaps at some point it leaked chemicals into the water, causing mutations. And maybe these mutations were of a more recessive variety, such that Rex's parents are normal (British) dogs, but he can talk due to the Punnet Squares aligning just right to finally activate his secret spaceship-fuel spill genes. Upon learning this, Rex's breeder would have most likely been terrified and put him out on the street, and from there he somehow ended up at the BB Bandits' base in London (it is, after all, the easiest city in the world to sell stolen artifacts in). From there, he quickly rose in the ranks as "being a dog" is basically the perfect disguise, and the rest is history.