Mastodon has a verification mechanism, though it looks different from Twitter's. We do not ask for your documents. Instead, if you have a website that you are known by, you can verify that you are the owner of that website.

Link to your Mastodon profile from the website, either through a visible link, or through a <link> tag in the head of the document, with a rel=me attribute, and link back to that page from your Mastodon profile. It should then get a green checkmark, like @Mastodon.