"John has a car and a motorbike" is a complete sentence. It's correct. The other example is incorrect: "a motorbike" is not a sentence but a noun phrase. Even if you were to write "John has a car and John has a motorbike" or "John has a car and he has a motorbike" (neither of these is good style or normal native-speaker spoken or written English), you wouldn't need a comma after "and" because both are short sentences and there's no problem reading and understanding them without the comma.
[EDIT: When I say that "the other example is incorrect", I mean that the comma isn't needed. The grammar isn't affected. The punctuation, in this case, tells the reader to pause while reading or speaking because "and a motorbike" is an afterthought and requires a slight pause. Contemporary ideas about punctuation would probably indicate an em-dash ("John has a car —— and a motorbike") or an ellipsis ("John has a car ... and a motorbike") to indicate the pause. That's just writing mechanics. It doesn't affect meaning, only timing and intonation.]