This is emulated as I'm sure the other videos are, but the PS1 back in the day had no way of running anything this crisp, so the emulator is `enhancing` it here. It's not an actual representation of what the game would have looked like.
CRT's smoothed out the image a little bit. Also, the screens were much smaller back in the day.
It doesn't really work right on "normal" PS1s yet, at least when it was making the rounds a few weeks ago, so you need either an emulator or modded/dev PS1 with more RAM to prevent crashes and most people won't have the latter https://www.reddit.com/r/psx/comments/1p45hrm/comment/nqjtdp.... Probably shared a few months to early.
But yeah, on a "real" PS1 it would be blockier due to lower res. The main rendering problems should be the same though.