Does a vet go through the same medical training as a doctor?

A veterinarian is a doctor, so in that respect they are the same. However, if you are asking what the difference between a human physician and a veterinarian, the major difference is in species treated. Human physicians (MD/DO degree) are legally only allowed to provide medical services to humans; veterinarians (DVM/VMD degree) are legally only allowed to provide medical services to non-human animals.