Oh, the shortcuts can sometimes really be in the way. I can’t count how often I pressed the numbers 1, 2 or 3 for point, edge or face selection while being “Object Mode” and having an object in “Local” view. After noticing that I was in the wrong mode, I tabbed into “Edit Mode” and tried to select vertices what I wasn’t able to do.
The solution: The object that I tried to edit was in a collection that I had turned invisible with my number pressing in “Object Mode” . But the display of objects in “Local View” isn’t affected by this contrary to the possibility of editing! Very confusing!
I solved this by holding CTRL and left clicking twice on the eye symbol of an invisible collection in order to turn all collections visible again (the first left click only made that invisible layer visible).
File is here.