Well, it seems that such sentence is originated in the Bible (Exodus 3:14)
I am that I am is a common English translation of the Hebrew phrase
אֶהְיֶה אֲשֶׁר אֶהְיֶה, ’ehyeh ’ăšer ’ehyeh ([ʔehˈje ʔaˈʃer ʔehˈje]) –
also "I am who I am", "I am what I am" or "I will be what I will be"
or even "I create what(ever) I create"
According to "I Am Who I Am: An Autobiography of God" by Rev. Thomas O’Donnell
Depending on your inflexion, your body language, and how the listener or the reader interprets them, those simple words can have so many different meanings. "I AM WHO I AM" - an arrogant, stand-offish, remote, distant comment that can be interpreted as "None of your bussiness who I am. I said it. You do it". Or "I AM WHO I AM" can have a softer edge to it, something like: "I'm sorry I can't articulate who I am any more clearly for you - I truly wish I could - but mortal, finite human words can't capture the essence of the infinite so let's just leave it at I AM WHO I AM. Sorry"
That been said, I have seen the video, the complete sentence is
You can beat me but I am who I am
Usually Stewie is an arrogant and stand-offish character with egomaniac tendences. He may play to be God in other chapters of the series. But in this occasion, what he is saying is that his father can beat him but that is not going to change who he is.
I'm not sure about the double meanings but I think that he may be talking about being gay, homosexual. His father can beat him but homosexuality can't be "cured" and less of all with violence.