Ubuntu's Snap Store can't update itself because it's running, so you get repeated prompts to update Snap Store that fail to do anything, or show errors. There's no indication that this is the case, or that the Software Update app you're using to update things is actually the same thing as Snap Store.
So, in typical Linux fashion there is no guidance on this and you have to crack open the terminal. I was able to piece this together from some Canonical Forum posts. You have to kill the Snap Store process, then update also via the command line (because if you update using the app, you've re-launched Snap Store...):
killall snap-store
sudo snap refresh
