Is it normal to have the home directory under git?
No, it is not normal for your home directory to be under git.

To remove just rm -rf .git

It is not a good idea to have it all under git as there are files like caches, trash that you do not want under a version control.
Theoretically, you could exclude them in .gitignore but I don't think this is practical and it makes it difficult for you to put other git projects in your home directory.

As to how it happened, we can't say. Something or someone made a git repo on your home directory.

For keeping configuration files under git look at the various git repos of dotfiles.