"My Mother Calls Me Teddy" makes no sense

Would Teddy really use the name 'Teddy' of himself, if that's where it comes from?

I mean, among friends, sure (but even then - why wouldn't he just let it be something that's ONLY between him and his mother?), maybe.

However, going undercover? Doing a drug deal? Helping a crazy lunatic murder people he thinks killed his wife?

I mean, Natalie knows him as 'Teddy', Jimmy Grants knows him as 'Teddy'. This name can be technically traced to him through his mom calling him 'Teddy', but he just keeps using it with these dangerous criminals and drug users/dealers/low-lives/etc.? (Is it 'low-lifes'? Sometimes plurals are hard to figure out)

What kind of sense does it make that you would take an endearing nickname your mother uses and then tell THAT EXACT NAME to some drug dealer so his psycho girlfriend and a crazy 'memory man' can use it to refer to you?

Wouldn't you come up with something completely imaginary that has no roots to your childhood and family history like that? Why would he EVER use 'Teddy' to refer to himself, if it's not only NOT made up by him, but something that can be traced to his mother, and consequently, him?

It makes NO sense that he would use this name, OR that his mother is where this name came from.

Either way, it makes no sense.