actually awesome and fast search engine (depending on which instance you use) with no trashy AI and ADs results also great for privacy, if you don’t know which instance to use go to https://searx.space/ and choose an instance closest to you
Of course it can be done, check your web server logs.
If you are using GET requests to send search queries to searxng, what you searched for will show up in the logs as
2024-10-31 123.321.0.100 /?query=kinky+furry+pictures
If you use POST requests the server admin can also easily enable logging those.
People hosting searxng can absolutely see what you searched for, along with your IP address, user agent string etc.