I know many people that want to be veterinary physicians one day.
That's understandable, they treat diseases, disorders, and injuries so it's a great profession.
how much schooling do they have to go through?
I'm not sure, although I think it is lengthy. In many countries veterinarian is a protected term, you can't use it unless you have a license.
Is there a penalty for using the term without a license?
I imagine that there is a strict legal penalty for defrauding like that. Activities of veterinarians like treatments are restricted to only veterinarians as well.