I had the same issue. I had changed my address everywhere with Apple I could think of but then noticed my old address under Billed to on my statements. I went to the Payment Info in Apple Music (as well as Books, App Store, and appleid.apple.com) with all the same result where I could not edit my billing address. In fact, on the web site, it showed me my old address under billing address but my new address under shipping. The billing section was not editable.
What I finally did was delete all my credit cards in my Account (note that you have to have at least one if you're the primary account in Family Sharing), and then go back and add them. Since the Apple Card had my new address for billing, it seems to have pulled that into my Account when I added that back. Note that I had to add the Apple Card back via the Apple Card in Wallet, where it says, "Make Default at Apple."
I hope that helps.