I bet Musk hacked something together (or has a column in TweetDeck if that's still around) that continuously searches for mentions. I wonder if there's a tool like that that covers more of the internet, although the primary users would only be famous people and/or their agents / social media staff.

There are lots of tools that do this if you are willing to pay for it. "Social Media Monitoring" or "Brand Monitoring" would be the keywords to search for