I just checked on an older desktop where I still have version 0.32.3 installed. No matter if I am on the public or pre-release channel, I get no notification when clicking on check for updates.
Edit: just checked on my 0.32 installation and inside of the anytype menu only about, settings and quick give a new window. the rest just closes the menu (even the hide option).
Unfortunately only MacOS and Windows are fully supported, Linux auto-update only works with AppImage, so we should add package to package managers or wait for the community to contribute it.