Translation:That is Doctor Wang's dog.
I imagine the term "veterinarian" has a different term, but could it be: "Na shi Wang, yisheng de gou" ? "This is Wang, doctor of dogs" ? I know it sound stupid but i wanted to know the right order
No, your question doesn't sound stupid, especially coming from European languages. I am taking Hebrew and Greek on Duolingo and word order trips me up quite often.
的 is the term to show possession, so it means 's. It separates Dr. Wang and dog, so it should be Dr. Wang's dog.
It doesn't mean "doctor of dogs" because the possession order is always the same. It means "That is Wang, dog of the doctor."