The case is arguably even more clear cut: copyright protects more-or-less exact copies. So copying old video games is not allowed. However, making something that is merely similar, but clearly different from the original, is not a copyright violation. For example, you are allowed to make a "Zelda clone" if you only copy broad game principles and vibes, but not specific art or level designs.
Generative AI mostly works by copying fuzzy styles instead of specific texts or images. There are some exceptions where models actually memorize specific material, but these seem to be relatively rare and probably require that the piece in question occurs frequently in the training data.
So in general, training on copyrighted material is probably legal as long as the model is not able to exactly reproduce the training data, while copying video game ROMs is clearly always illegal.
Of course, whether these things are morally okay or not is a different question...
Edit: Of course, to train on copyrighted material, you have to download it first. If you don't pay for the copies, this is arguably still illegal, even if the resulting model doesn't distribute any copies! (An exception might be content that is directly embedded in websites, because copying websites into the browser cache is allowed, even if they are under copyright protection.)