All the "App Store" apps like Discover are merely frontends for your system's underlying package manager (apt for Debian and derivatives, dnf for Fedora and its derivatives).
The underlying package manager does the updating of packages: if you've installed it through the package manager (which is usually most stuff on an install) - it'll get updated.
Discover just gives you a nice, user-friendly way of interfacing with the package manager(s) on your system so you don't need to bother with the CLI if you don't want to (that's what "frontend" means - a nice, friendly UI for underlying services).
And yes, you can have multiple - for example apt and Flatpak. Discover and friends should update all.