Does this require using apple specific software though? I doubt it would work if I have an app/program installed on o ly one of the devices.

Handoff can work with third party applications though mostly it’s Apple stuff. One notable exception is you can hand off webpages from your phone to macOS and it’ll be handled by any browser.

Any cloud syncing service will sync files the second you save them. IM clients sync by their nature. Web browsers have had sync capabilities for years.

I've lived in the Apple ecosystem for a few years and Handoff/Continuity can do some impressive things (like the webcam thing of late, or cross device phone calls) that I ended up not using in practice. That's just me though. I prefer the simple way.

When Handoff or associated features don't work when they should, good luck troubleshooting.