edit: Thanks for all your answers so far.
I have decided to develop my own solution, both because it is fun and I can then form it exactly as I want to. Once I am done (which might take some time, because I first have to learn some Python for it^^), I might share it on github if it turns out well enough (will edit here with a link, but don't expect anything for at least a few months).
As mentioned in the answers and comments, people might find this question through google and your answers might help them, so I will not accept one answer now but instead leave everyone the chance to add other tools they know about.
I was just recently thinking about how to best organize exercises to later reuse them. Sometimes you have a nice idea or see something nice (online/in a book), but you are not teaching a fitting class at that moment. Or maybe you just want to reuse some exercises (maybe with altered numbers) when teaching the same course again a few years later.
I am especially thinking about the case where there is no textbook containing exercises, but instead the instructor has to come up with questions themselves (themself?^^).
Yes, of course you can simply write them down somewhere, but wouldn't it be nice to have a database where you simply search for "Gauss algorithm" to get all exercises for that, or where you filter by "Analysis, week 5" tag to show you all exercises you had on this worksheet the last few times you read this course?
I was thinking of maybe coding up my own solution for the problem, but before investing time into that I just wanted to check if someone here knows of a good (free/open source) tool for that; if possible also with LaTeX support.