Regex Version of strip()
Pre-compilation

In theory, you're going to want to call this function more than once. That means that you only want to pay the cost of regex compilation once, and you should move your re.compile calls out of the function, setting your regex variables in the module's global scope.

Type-hinting

s is s: str, and char is (I think) also char: str.

'Removed from the string'?

I think this is the fault of unclear requirements, but - it would make more sense for the char argument to be the character[s] to strip from the string edges, not characters to remove from anywhere in the string. As such, you would need to re-evaluate how you create your regex.

Combine left and right

There's no need for two regexes. You can use one with a capturing group:

^\s*(.*?)\s*$