Older version of "Ubuntu Software" is still in my system

Solution 1:

These are not "old" and "new" app. Both are present in Ubuntu 20.04, there are two different applications that can work in parallel.

There are two basic application formats that can be installed on Ubuntu: .deb packages, which are native Debian/Ubuntu format, and snap applications, which are an attempt to create some universal application format independent from particular Linux distribution. There are also two different sources for these applications: .deb packages are installed from Ubuntu repositories, and snap applicationas are usually installed from Snapcraft server, https://snapcraft.io/ .

The application called "Ubuntu Software" (also known as "Snap Store") is focused mostly on snap applications. It is slow "by design", because of reading the application list everytime from Snapcraft server. The other, called "Software" (a.k.a. "GNOME Software") is focused mainly on .deb packages from Ubuntu repositories, however it also features snap applications, but not so many as Snap Store.

There is no "Software Center" in Ubuntu 20.04.

Look at this screenshot (it displays an older version of Ubuntu Software aka Snap Store, when both applications were 3.36.1).