Is it normal that my baby bristlenose pleco has almost no tail?
Nope. And that's not really a baby anymore, that looks more like a juvenile, probably a few months old at least. But definitely a deformity.
Do you think he was born that way, or maybe it was nipped by other fish in the store?
Hard to say, it should grow back if it was nipped off. Do you know what kind of fish he was kept with in the store?
I would say he was born that way, as it doesn't look like nibbling damage. He may never regrow or even grow the fin.
Looked like there were only albino bristlenose the same size.
Separate names with a comma.