As for this, it might be that you have a different sketchbook setting in each of the Arduino IDEs. Arduino IDE 2.x has a different preferences storage system than the classic Arduino IDE, so any custom settings you configured in the classic Arduino IDE preferences are not automatically transferred to Arduino IDE 2.x.
Libraries are installed to your sketchbook folder, so if you have a different sketchbook folder then the libraries installed by one IDE will not be available in the other.
You can check the sketchbook folder location by opening File > Preferences from the Arduino IDE menus and then checking the "Sketchbook location" setting.