IMO the reason to suspect that git is independent from the SaaS products, is that you can create and commit (and merge and most other things) to a local repo without ever pushing it to GitHub or any other site.
It's baffling because practically all learning resources on git emphasize its decentralized nature. Even the online free book explains that several times. They never left me with any confusion regarding the difference between git and github, even though my first version control was the centralized subversion. I don't understand how any developer can learn git without this idea being drilled in constantly.
If you didn't already know git was independent of the SaaS products, you'd have no reason to suspect it was independent.