Are Lutherans the only branch of Christianity that does?
No.
I go to an Episcopal church. There are Presbyterian and Methodist churches that are "open and affirming," and of course there's UCC, MCC, UU (well, they're vaguely Christian, at least some of them). I don't think it's that useful to classify by denomination; usually it depends on the specific church.
Also, when you say, "Lutheran" I assume you mean ELCA, not any particular Lutheran branch in, say, Sweeden or Germany, not to mention the quite conservative Wisconsin and Missouri Synods in the U.S. "Lutheran" is hardly a specific designation.
Ty Alyssa! It is so good to hear what you say!