Play PS4 games with a different medium?

I have to briefly explain the situation:

Almost two years ago I borrowed several PS4 games from my cousin, but now she wants to play the games herself again. Now to my question: If I give her the discs now, then I can no longer play the games. If I were to buy the discs of the games myself, could I play the games that were already installed? Because then I would no longer play the games with the 'original' disc with which the games were installed. Are there any problems with the copyright… The games? LG

It has absolutely nothing to do with copyright law?

And yes, that should work.

I thought there might be problems because it is a different disc.

That is definitely possible. The game is on your hard drive. The CD is only used to confirm that you own this game. If you bought the game online, you could start playing immediately without reinstalling it.