who am I
from: How I Met Your Mother
Robin : I totally wanted to come. I got stuck at work again. I feel like I live there.
Sorry, I missed your party again.

Ted: Hey, ain't no thing, but a chicken wing, mamacita.
Who am I?

Robin; I guess there's no chance your two-dayer turn into a three-dayer.

What does "Who am I?" mean?
I have no idea what I am doing?
or any other connotations?

Question

who am I
from: How I Met Your Mother
Robin : I totally wanted to come. I got stuck at work again. I feel like I live there.
Sorry, I missed your party again.

Ted: Hey, ain't no thing, but a chicken wing, mamacita.
Who am I?

Robin; I guess there's no chance your two-dayer turn into a three-dayer.

What does "Who am I?" mean?
I  have no idea what I am doing?
or any other connotations?

Answer

Probably it means that his previous line is an "impression" - he is mimmicking someone else, and he wants Robin to guess who.  If you scroll down in this link, you'll see an explanation of "impression" in this use. https://dictionary.cambridge.org/dictionary/english/impression