Currently, when I want to update my flatpaks I have to open the gnome software center and look for updates there. My other software (apt, snaps) are updated by the "Software Updater", aka update-manager. Is it possible to also update flatpaks via the Software Updater?
No:
update-managercurrently handles only deb packages.It is an Open Source application, so any community member can fork the code and/or can contribute additional features.
A proper Flatpak/Flathub (and Snapd) install already includes automatic updates, so it's understandable why perhaps there hasn't been a lot of community interest in contributing this feature to
update-manager.