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.
I'm sure the actions of veterinarians are under heavy observation.
Yes I agree. In some jurisdictions, other professionals can perform some animal treatment, either through exemptions in law or lack of prohibitive laws.
There are always exceptions to any rule.
You're right, veterinarians are also usually assisted by paraveterinary workers like technicians and vet assistants.