This site claims to run Mastodon but its front-end looks fresh and smooth!

That instance just has a custom homepage written in React. The actual social UI is normal Mastodon with some color tweaks.

It is possible to use different frontends for Mastodon, though. Web apps like Elk can either run on a dedicated server to log into any Mastodon instance, or can be hosted by instance admins as alternative frontends.

If you look at any profile page, it's the classic Mastodon layout: https://veganism.social/@mascottus

The probably just customized the home page

If you really want something a bit less austere than Mastodon you can have a look at https://iceshrimp.social/