I bought New Horizons digitally back in April. I don't remember seeing anything about a code, though. Where do I get my code, if it doesn't expire?
EDIT: ...So that's what that email was about! Good thing I didn't delete it. It says you can't use it while you have an active membership, but I can use it any time, apparently. I'm glad I saw this thread, or I might not have looked through my old emails for the code!
I burned through a month's worth of free trials when the game launched because you can get them on My Nintendo as well. The My Nintendo ones do expire, but I keep seeing more put up, possibly one a month?
I had one from buying ACNH on the eShop, one from this promo that I may have been able to redeem twice somehow, and then two My Nintendo codes.