Game design majors are a pretty niche major - as the industry grows (this is my field) more of these programs are popping up, but they’re relatively new. Therefore, the requirements for the field aren’t as solidified.
I think your portfolio is going to depend on what you want to focus on and what kind of game design major the program offers. Majors that are heavier on the coding and development aspects of game design are probably going to prefer you having a game or game features to showcase - or rather, you’d be more competitive if that’s what you had.
The NYU Game Center’s BFA program explicitly says the creative portfolio can be “a game you’ve made (digital or non-digital), a computer program you’ve written, a film you created, a piece of music, a creative writing sample, a portfolio of artwork, visual design, sculptures, photography or other creative pieces. We are looking for your talent and creativity in whichever medium you choose, so choose an example that you think best shows off your strengths and ideas.”
So I’d think about what your specialty is and where you can really show off your strengths. Are you a talented character designer? Do you do environmental artwork? I work at a very large video game company and our artists tackle different kinds of art - as an artist at an Ubisoft or Naughty Dog, you may work only on environments or only on characters or only on [insert element here]. But at a smaller, indie studio, you may be one of only a few artists, so you may work on more. So focus on what your strengths are, in whatever medium that strength is.
BTW, the NYU application is really cool. I LOVE that they make you write a short essay on a game that’s not a video game, like a tabletop game or board game. So much about good game mechanics can be learned from studying great board games, and many video game creators (including myself) are major board gamers. After work and on lunch breaks you can see people around here playing D&D, Magic, Warhammer, and all manner of other games. Sometimes when talking about video games people can get distracted by the shiny, but talking about analog games makes you really think about the mechanics.
RPI’s program seems to have different concentrations based on different disciplines in the industry (I love that they have a cognitive psychology concentration! That’s what I do in games 
) so I’d tailor your portfolio to what you want to do there. You wouldn’t want to submit a strictly art/illustration portfolio if you had interest in the CS concentration, you know?
Both of these programs look really awesome. Good luck OP. Make sure you are also considering more traditional universities; most people who get into game design don’t major in game design: majoring in art + computer science is a perfectly good way to get into the industry, too. Also check out USC, as they have strengths in game design as well.