Actually I’ve seen people mention it since the first episode.
It’s Capaldi’s wedding ring. He never takes it off, even when acting.
To add to the story, he refuses to take it off because when his acting career was struggling, his wife never gave up on him. When he landed his first major gig, he decided to not take it off, to represent he was there because of his wife’s support.
I figured it was something like that, but working it into the story, the doctor is married and maybe the ring helps keep him from flirting with his companions. He mentioned something about stopping that, I think.
Friendly reminder that anyone born between 1985-1998 didn’t get their hogwarts letter because Voldemort’s ministry wiped out the record of muggleborns
additional note that you would likely only get a letter to Hogwarts if you live in or in a country near Britain, as shown in book 4 there are, in fact, other schools. The U.S. probably has its own wizarding school, same with India, probably Brazil, and a few dozen other places around the world. (ministry thing still applies, of course, pretty sure the ministry of magic is a global thing)