Word for "An expression or saying one uses so much that it comes to define them" [duplicate]
Catchphrase would fit the first sentence.
I don't see how the word you ask for in the first sentence would fit the second sentence.
Shtick
one's special trait, interest, or activity : BAG
He's alive and well and now doing his shtick out in Hollywood. — Robert Daley
[Merriam-Webster]