No, it’s not „always up“.

There are three main ways how Google, Bing,… can track you:

1. When you’re doing a search while being logged in, it’s probably you
2. If you’re not logged in, they can set a cookie to recognize you on your next visit (although they may not be able to link this to you, your email address,… but that’s not needed). They may mix your searches with those of the other users of your PC, when those are using the same PC, browser and account (e.g. if you have a family PC with a single windows/Linux account that everyone uses)
3. Even if you’re not logged in and don’t accept / delete your cookies, they still see your IP. Depending on your ISP you might have the same Ip for a long time or you might have it rotated regularly. Now they could only track the searches of your household (assuming everyone isn’t logging in and deleting cookies immediately)

With Searxng, they can only do the last variant. But assuming you use a “real” server in the internet (and not one at home), it will likely have the same IP for its lifetime. And if you’re using it alone, that’s the only thing they need to identify you and track your searches. The more other people use your instance, the less useful this kind of tracking gets. Too much noise to identify a single person.