If you have two versions of SketchUp open, or two instances of 2023.1, pasting into the second instance may not work at first. To work around that, do a Copy of anything in the second instance. That seems to wake up the clipboard, and you can now copy and paste from the first instance into the second instance. After having done one Copy action, copy and paste should work until you next restart SketchUp.
I added this note, to say that the 2023.1.3 update fixed the problem with copy and paste.