Mastodon doesn't have the same concept of verified accounts that Twitter does (or did until radical changes to Twitter's verification system were announced piecemeal on Twitter in early November 20221)).
Anyone with the verified checkmark in their display name has added it as an emoji. It has no functional meaning.
You can add verified links to websites controlled by you:
etc.
To do this, you must add a specially-formatted link to your Mastodon profile on the page you wish to verify.
You should trust the URL verification only as much as you trust the user's instance to carry out the verification. For large or familiar instances, that might be quite a lot. Perhaps less for instances you don't know well or at all. This is more complicated than monolithic social media, where a single authority may grant verified status to an account (but often based on an opaque process).