Theoretically, this site can support any font that it is licensed to use and can distribute to users in a reasonable way. For most websites, this means using common fonts that come with operating systems (e.g. Arial) or cacheing fonts from an API like Google Fonts. Relevant thread if this interests you.
However, if your concern is with your username, none of this will be helpful to you. The website is coded to use particular fonts, and as users, we can't change those fonts (unless you have a browser extension of some kind, but that's not the same). If the site is working normally, your username will always appear in the same font -- and if not, another prescribed one that works as a backup.
The reason some people's usernames appear differently is that they use special characters and symbols, which you will not find on a standard keyboard. They are referenced through Unicode, which is a standard for the different symbols you see within computer text. Many different Unicode characters exist for each letter of the Latin alphabet, and people use them to style their usernames, so to speak. You can find text generators online that will 'convert' your name to have a Unicode style, or you can hand-pick symbols to use yourself. Here's one good reference for that.
Fonts come back into play because they are responsible for the appearance of particular characters. Not every font supports every Unicode character, so maybe the site will use a backup that supports the characters in your username... but it's not really the same. And anyway, you'd generally just get a common sans-serif font like Arial or Calibri that is specifically designed for compatibility.
Sorry if this was confusing, tl;dr you can't change fonts but you can use special characters to make it look like you did.
