Searx is not a search engine with its own index, but rather, it is a meta-search engine, and operates by forwarding your query to a customizable list of other search engines (like Google, DDG, Wikipedia, etc) and gives you aggregated results. It is more privacy respecting, because the search providers cant run any tracking JS, see your IP, give you tracking links, etc. People may have their own instance for any number of reasons, including: hosting in different places to get different geo-specific results, to be faster by being closer to the end user, to change default themes/settings, to reduce the load on other instances. Nowadays, the fork SearXNG is more popular than the original.

https://searx.space indexes most of the public instances.

it's a meta-search engine (a search engine that searches through other engines). I self-host one myself (although I host SearxNG, a fork). You could argue you can be identified because of the same IP, so you shouldn't self-host for yourself only - but there's very real legal and practical limits to tracking without cookies, so I'm reasonably fine with this, especially since I'm not the only one using my instance (although the user count is under 10).