Belonging to the infamous generation from which these terms appear to come I don't think "my bad" and "my bag" are the same. My bad, as you have pointed out, means "my mistake". I really don't know where it came from. But "my bag" means, as far as I know, "something I like" or "something with which I'm associated." In the movie Austin Powers, when Austin arrives in the present after being cryogenically frozen he is given a box of his belongings. Contained therein is a Swedish [you know what] Enlarger. Somewhat flustered, Austin tells his new partner, a young lady of the '90s, that it's "not my bag, baby." Austin is supposed to be from the 1960's so I would say that "my bag" was a '60s UK term.