Yes, Git does rely on a POSIX system, but be advised that POSIX specifies a lot of things: a shell environment (i.e. command line environment), operating system interfaces (i.e. libraries that implement file I/O, threading, etc.), regular expressions, directory structure (i.e. /dev/null, /tmp), and utilities.
Given how I've defined POSIX, it should follow that Git relies on practically all of POSIX. However, if you read the Git developers' CodingGuidelines, you'll see that Git uses POSIX as a means of implementing wide portability.
msysGit gets around the compatibility problems because it is a POSIX environment.
Running Git on Windows generally means running on top of a POSIX run-time environment. This is the basis for Git for Windows. You first start Git BASH, and in that environment you run Git commands.
I can assure you that Git works fine in this manner!
If POSIX and the command line make you hesitant, you may want to explore Git UIs like Atlassian SourceTree or GitHub for Windows.