60 fps games better on 120 fps TV / monitor?

Hi, I wanted to ask whether games that support max. 60 fps run better if, for example, 120fps mode is set on the Xbox Series X or PS5 (the TV also supports it)? Is there an interframe calculation on the consoles or something or do the games have to support 120 fps natively?

I would say the game has to support it too. Gladly informs me if I'm wrong.