If you disapprove of the overly submissive way someone is acting — like the teacher's pet or a celebrity's assistant — call them by the formal adjective obsequious.

There are many words in the English language for a person or an action that is overly obedient and submissive. Obsequious people are usually not being genuine; they resort to flattery and other fawning ways to stay in the good graces of authority figures. An obsequious person can be called a bootlicker, a brownnoser or a toady. You can also say that someone gives an obsequious bow, a gesture that means, "your wish is my command."

Definitions of obsequious

adj attempting to win favor from influential people by flattery

bootlicking, fawning, sycophantic, toadyish
lacking sincerity

adj attentive in an ingratiating or servile manner

obsequious shop assistants”
submissive or fawning in attitude or behavior