Not enough space on disk to install App Store app, but which one?

Check the "purgeable" space of your APFS disk with Disk Utilities. It seems Finder count this space in free space, but not App Store.

But you can trick macOS to clean up APFS by creating a huge garbage file, then deleting it.

You can create a huge file that will force macOS to clean purgeable files to free you space. In order to do so, type this command in a terminal:

dd if=/dev/zero of=~/hugefile bs=15m

It will create a file called hugefile in your home folder, which you can check the size with Get Info and stop when it's big enough for you, using ControlC. Or you can simply let it run until you are out of space in the disk and things start to stop working.

This command takes a long time to allocate the memory, you can also stop it when it's 5~10GB.

In practical, what I have done earlier is letting dd run for about 30 seconds, then killing it with ctrl+c, and deleting the file:

rm ~/hugefile

Hope this will help in your case too. Then App Store happily installed the update if it is trying to udpate.