I run HTTP Toolkit (
https://httptoolkit.com) which passed $2k a couple of years back. No longer a side project, as it's made enough money for me to work on it full time for a fair while now, but it certainly started that way, and it's still a one-man show (plus many wonderful open-source contributors).
I suspect that'll be a common theme in answers here though: if you have a side project making $2k a month, in most of the world that's enough for you to go full-time and try to take it further. If you can make $2k/month on something working only part-time, you can definitely make a lot more if you focus on it.
On your questions: HTTP Toolkit is a desktop app (plus a mobile app and other components for integrations) but it's an Electron app that effectively functions as a SaaS (with a freemium subscription model) that just happens to have a component that runs on your computer. And actually getting to $2k wasn't overnight at all - it took a couple of years of slow steady slog. A few inflection points that made a notable difference (releasing rewriting support & Android support particularly) but mostly it was a matter of "just keep pushing", trusting the trajectory would keep going, and steadily grinding upwards. It's great where it is now, but it's hard work - a solo business is not for the faint of heart!