If you are sharing the same $HOME, then it is up to you to make sure that works as expected. If both systems are using the same desktop environment and you have different versions of extensions etc., then that will indeed cause problems. A simple workaround is to use a different desktiop environment on the two systems so the settings don't overlap.

Although it is indeed common to share $HOMEs across distributions, that was much simpler a few years ago when there were fewer desktop environments around and most settings were about the shell and not the GUI. You can still do it, but you do want to be a bit more careful and try to avoid situations where the two systems have different config options. So install the same themes on both systems (this should happen automatically if they both have the same GUI and version), make sure you install the fonts on both systems and so on. As long as you keep them more or less in sync, you should be able to minimize the issues.

To make it 100% robust, however, you will need to have the configurations separate. One trick would be to move your config files and directories to another place and then symlink them depending on what system you load. For instance, you can have a partition that is only mounted by Ubuntu and one that is only mounted by Debian and store your ~/.config/ and various ~/.foo files there. Then, in your $HOME, you replace all these with symlinks to /mnt/dynamicallyMountedDrive. You set up the two systems to mount either one or the other partition to /mnt/dynamicallyMountedDrive and that means your symlinks will always point to the right setup for the right OS.

This way, you still have your actual data shared between the two systems, but the configurations are separate.