I remember playing a Prince of Persia game on PC where the prince wore a red shirt or something like that. It looked similar to your pictures but instead of wearing white, he was wearing red. I was never able to beat the game on my own, there was a part where a wooden bridge was constantly falling and there was also what I believe was an immortal skeleton.
That part of the game was impossible for me but there were older family members that did complete the game while I watched in amazement hahahaha
Another old game from those years was Wolfenstein 3D, did you play this one?
I always saw the prince dressed in those white LOL pajamas. I never got to play Wolfstein, but I could try, I love retro games.