Somebody who considers themself so tied to someone else that they are only there to do the other's bidding. A slave. By analogy with an actual third arm, with which the owner of the arm could hope to do so much more, but which has no independent life of its own.
Mary is trying to treat Jimmy like her third arm again. She won't even let him go out for a night with the lads.
The arm that ends up in the middle while cuddling.
My third arm is in the way.