The nicest way to use Mastodon on a computer is probably through your server's website.

Your server's website address will usually be the same as your server's name, for example mastodon.coffee's website is at https://mastodon.coffee

You can log in there using the same ID you use on the app.

However, you can only log in on your own server's website. You cannot log in on another server's site because each server is independent. (It would be like trying to log in on Yahoo Mail with a GMail ID)