How can I remove this alert badge from my settings icon?
It's a bug in recent version(s?) of iOS. The fix is to log out of iCloud and log back in. I have the badge as well, but haven't tried the fix yet.
From reddit:

So on a hunch, I signed out of iCloud and back in. Badge gone!
When I signed back in, there was a prompt about not having backed up in over 7 days. So maybe there is a bug that's causing that to lurk under the hood somewhere.