Learn English – Is “being on the bats” an expression

Your thing is what drives, motivates, interests you most. Something you are usually good at, your main focus. Often an area of expertise, after a while.
We may assume that gorillas were Dian Fossey's "thing" and Playboy Bunnies are Hugh Hefner's "thing".

It can also be your "modus operandi", a behavioural pattern or quirk.
In this sense, Charlie Chaplin's waddling walk was "his thing". Charles Ponzi even had "his thing" named after him.

"Teachers pet" is a stand-alone idiomatic expression, and I have never heard "pet" used this way with other things. I suppose it could be, but it doesn't sound very clever.

In creative writing, of course, you can metaphorically say someone is a "pet":

He was her pet, her plaything, something to whistle over when she was lonely and to brush away when she became bored.

but this negative meaning of an owned animal is quite different from "teacher's pet", which is usually more positive.

I suppose "teacher's pet" can apply to adults, but it would be weird, with possible sexual overtones. In this case you would soften the idiom with "like a teacher's pet" to diminish any awkward connotations.