Ubuntu Software doesn't work and why are there two Software Center in Ubuntu 20.04?

This is because the pre-installed Ubuntu Software is a snap package (source: article on OMG! Ubuntu). You can check this by running the following command:

snap list

You should see an entry called snap-store which corresponds to the pre-installed Ubuntu Software.
You are supposed to be able to run the pre-installed Ubuntu Software by running the snap-store command in Terminal. If it fails, it should show some useful error messages.

Secondly, the pending updates shown in the GUI application are updates for two pre-installed snap packages. That's why they don't appear in the list of packages that can be upgradable by apt.