PSA: The "verified" symbol you see on Twitter etc means absolutely nothing on Mastodon or the Fediverse. On here, symbols like 
are just custom emoji that anyone can use and put after their name. Don't be fooled! 😁
There are ways to verify your identity on here, but the symbol is not one of them.
The easiest way to verify your identity is if you have a website that people already know is yours. You can paste a special code into your website which will verify your address and it will turn green on your Mastodon profile (see here for instructions: https://mstdn.social/@feditips/106274105538953001).
Alternatively, you can set up a server as a subdomain of your known website address. For example, the European union set up their own Mastodon server at https://social.network.europa.eu, and we know it really is the EU because the server address is part of the EU's well-known europa.eu domain.
p.s. If you want to see an example of a verified link on Mastodon, try looking at the profile of @libreoffice
Unfortunately the verified links aren't visible on the official apps yet, so you would need to do this via the website or a third party app.
I've included a screenshot so you can see what I mean.
@feditips @libreoffice I have yet to work out how to get the link in my YouTube profile to get it verified…
@feditips @libreoffice it’s wrong though, I don’t own me.uk. I own www.me.uk which is different. Mastodon gets it wrong taking the www off the front. I do however have an even shorter web address than me.uk though, several actually, E.g. 4.gg
Ah, think you have found a bug there!
I guess not many people have a domain that's www.something.
@feditips @libreoffice I would suggest not taking off the www anyway. People can list the site without www if that works for their site (obviously does not for www.me.uk). Even with a more usually domain the www prefixed version does not have to be the same page as without.
@revk @libreoffice
I am not sure if YouTube accepts HTML code so it might not be possible?
By the way, that's an amazingly short website address! 😮 Shortest possible?