Actually, this idiom does feature a word craw, not claw. A craw is a part of the digestive system of a bird or some kinds of animals or insects. So if something sticks in your craw, it's like saying that you can't stomach it or it's not acceptable to you.
Why some English speakers people say sticks in my claw (or his claw, her claw, etc.) might have something to do with the well-known fable of Androcles and the lion. Obviously, if something were sticking in your "claw" like the thorn that was sticking in the paw of the lion, it would be something you’d have a hard time living with too and would want it out of there tout de suite.