As a nitpick about language and terms, I think it's important to recognize that Mastodon and Lemmy, Calckey, Friendica, Akkoma, kbin, PeerTube, PixelFed, etc. are not websites. They're different server types, which come with a web-based front end. These servers run thousands of websites, each of which is its own social networking site.
This is actually an important piece of how the Fediverse fundamentally works. Even if we're not using different server types (Lemmy, Mastodon, etc), we're still largely using different websites.
And while you can install an iFrame or whatnot in a website to serve content from other websites, that's generally not what happens here. Here, you view content on whatever website you've logged into. This means copies of posts are being sent all over the network, and are stored on and hosted from each site that receives them.
What you read is being served from the site you're using. When you post something, that post gets copied to other sites, and other people read those copies.
A common mental model I see people have in the microblogging corner of the Fediverse, following the Twitter migrations, is that "Mastodon" lives out there in a singular place, and people just access it from whatever website they registered on, like it's a mainframe and they're using a terminal to access it. But this is much more like a bunch of independent web forms (or blogs, in the microblog case) where people are actively cross-posting posts and replies, creating copies of discussions.
There is a whole range of alternatives, several for microblogging (Misskey, Calckey, Pleroma, Akkoma, ...), some for macroblogging (Friendica, Hubzilla) and some that specialise in certain functions (e.g. Pixelfed for pictures, Bookwyrm for books, etc.).
Some short descriptions can be found here in the JoinFediverseWiki: https://joinfediverse.wiki/What_are_Fediverse_projects%3F
Its kind of both, it has its own centralized website, but also self-hosted instances. They were popular with web designers before social media took over the internet because they were easy to set up and super customizable.