Just like you don’t have to use git with GitHub, you don’t have to use Flatpak with Flathub or your distribution’s app store. You can host Flatpak repositories yourself.
This is an excellent short tutorial showing you how: https://blogs.gnome.org/alexl/2017/02/10/maintaining-a-flatpak-repository/
* Note: if you get an error about the freedesktop SDK or platform not being installed, you’ll have to install them manually. See https://docs.flatpak.org/en/latest/first-build.html
* Also, your flatpak install command will require sudo when run with a remote added with flatpak remote-add --no-gpg-verify (don’t do this in production, it’s just part of the tutorial to get you started, after which the tutorial shows you how to generate GPG keys, etc.)