Not necessarily. A lot of video software either leverages the Windows/MacOS system codecs (ex. Media Player Classic, Quicktime) or proprietary vendor codecs (Adobe/Blackmagic).

Linux doesn't really have a system codec API though so any Linux video software you see (ex. VLC, Handbrake) is almost certainly using ffmpeg under the hood (or its foundation, libavcodec).

FFMpeg is definitely fairly ubiquitous, but you are overstating its universality quite a bit. There are alternatives that utilize Windows/macOS's native media frameworks, proprietary software that utilizes bespoke frameworks, and libraries that function independently of ffmpeg that offer similar functionality.

That being said, if you put down a pie chart of media frameworks (especially for transcoding or muxing), ffmpeg would have a significant share of that pie.

Brings back memories. There was a time when the fork, libav, became the default on Ubuntu, and ffmpeg commands would say "this command is no longer maintained" or so. That was where I learned that there was a fork, and I thought ffmpeg was going to die as a result because there was heavy development activity on libav compared to ffmpeg initially. Surprise, ffmpeg outlived its fork!

This post talks about the situation back then: https://blog.pkh.me/p/13-the-ffmpeg-libav-situation.html