I was trying out Same Game for the Super Famicom today. It just got sold and I had to send it. And it dawned on me why it didn't look like in the video.
I had tried it out a while ago, and searched for a setting to get Mario heads but couldn't find it. Most likely it is that small cart that changes what symbols the games uses.
I don't think it is a game of skill. You just try your luck and if you fail you use the option to back up a couple of moves. I tried different ways but couldn't find any way to actually figure out how to solve the puzzles. It is just trial and error. The puzzles aren't even levels, they are just randomly created. You do get a high score, and that seems to be saved. I beat one of the high scores, so it is possible there is some skill involved. But maybe I was just lucky.