So, if you don't want to set up your own Mastodon instance under your domain, you can instead link your Mastodon account to your homepage running on a domain of your choice. It's not something I bother to do, because I'm nobody important, but if I was e.g. Taylor Swift or something, I could show that my Mastodon account was affiliated with www.taylorswift.com, and thus give you a pretty good idea that I'm actually the real deal.
From the Edit Profile page on Mastodon (v3.5.3):
> You can verify yourself as the owner of the links in your profile metadata. For that, the linked website must contain a link back to your Mastodon profile. The link back must have a rel="me" attribute. The text content of the link does not matter.