I believe Baloo (a KDE project) does exactly that (indexing local content). There are different tools you can then use to search for indexed data: I assume KRunner will be the closest to Spotlight, but you can search for files right from Dolphin, if you use it.
To be honest, I've found Baloo to be a resource hog in the past, so I've now the habit of disabling it right after installing a distro which comes with it (eg. Kubuntu) or not installing at all otherwise (eg. on Debian or Arch). I should probably give it a second chance, though: on paper, that's the right approach to search.
Baloo ist basically Spotlight and probably more but dare you are not that typical happy smile buisness user - have a few kernel source tree's laying around in your /home and you'll likely are getting friends with Baloo sooner (spinning rust) or later (ssd) because that beast manages to saturate even SSD IOPs and while doing so hands out segfaults on every other file.
You can switch to disable content-indexing and just search for filenames (80% rule) but it's hidden in systemsettings or you even need to go to balooctl - also index corruption is a thing...
That beeing said, if you are a brave KDE user and don't mind the hassle it might even work 80% of the time - in theory it's great idea but it never worked reliable for me - if you are a bored dev there is probably lot's of low hanging fruit there - seccomb, some simple heuristic to not overload on IOPs, there are probably more efficient db structures than LevelDB and so on...
It's step forward from nepomuk? that did the full rdf semenatic web stuff and fed a relational database and killed your hdd-based desktop reliable in the early 2000ies but it's still a nasty surprise when using KDE.
It's still a cool idea but it needs some love and contributors to work reliable on all kinds of nasty setups.
I’ve found searching for files on KDE (with Baloo) to be, for some reason, really inaccurate and hit-or-miss. Sometimes, even if I remember the exact name of the file, it won’t show up in the search results. Partial names have a lower probability of showing up. Slightly inaccurate names (with a Levenshtein distance of 1 or 2 to the actual name) have a very low probability of showing up. I end up searching for it using find or ag (the silver searcher).
Baloo is also indeed a resource hog. It can use 100% of a CPU core for hours and hours, while it’s indexing. But I’d be fine/happy with that if it just worked properly.
Baloo is alphabetically first dependency of Dolphin in Debian (both Stretch and Buster), so either you have to use a different file manager or disable it after install.
To be honest, I've found Baloo to be a resource hog in the past, so I've now the habit of disabling it right after installing a distro which comes with it (eg. Kubuntu) or not installing at all otherwise (eg. on Debian or Arch). I should probably give it a second chance, though: on paper, that's the right approach to search.