Non-arbitrary decks of cards, or Uno with Set cards

So this is something I thought about a while ago.

A deck of cards has 13 numbers + face cards, and 4 suits.  This seems a bit arbitrary to me.  I thought to myself, we can do better than this.

How about something like, 3 numbers, 3 suits, and 3 colors?  This gives you 3^3=27 cards, which I think is much more pleasing.  Also this could be considered a "3 dimensional" deck, as each card could be uniquely placed in a cube by treating each value of each attribute as a distance from a cube edge.

Or, you could make a 4 dimensional deck, with 4 numbers, 4 suits, 4 colors, and, I dunno, 4 shadings. 4^4 = 256 cards.

Well, after thinking about this, I re-stumbled upon  a game I had played a while before that, which I now suppose could have at least unconsciously led to my idea in the first place.  This is a card game called Set, (the family game of visual perception it says on the box.)  This game has the 4 numbers, 4 shapes, 4 colors, and 4 shadings.  There are 3 of each instead of 4 of each, giving the maybe less "mathematically pleasing", but maybe easier to work with, 3^4 = 81 cards.

The game they designed for it is fun I  guess, I guess it's gotten awards and stuff.  It is mostly just about being faster than the other person at recognizing patterns.  But anyway, I bought a set and set to thinking about my old idea of using these cards to play actual games.  The game I started with worked pretty well I think.  That game is Uno, or I guess Crazy Eights.  The rules are, as I have tried out,

-Each player starts with 7 cards.
-A card is drawn from a draw pile
-Now players take turns putting down cards that "match" previous cards on a growing pile.
-"Matching" cards have either two OR three things in common.
-Two things in common means it's the next person's turn to go
-Three things in common means you can go another time if you choose to (my fiancee says you are REQUIRED to go a second time, which I guess makes things more challenging, but also maybe a bit more complicated...)
-If you have less than two things in common with the card already down, you have to draw until you can play again.
-The first person to get rid of all their cards wins.

This is basically the most basic rules of Uno but with more than just an arbitrary number of colors and numbers.  As my fiancee points out, nobody likes you when you call someone out for not saying "Uno", (wasn't me!)  so I leave it up to the prospective players whether or not to add a rule requiring the calling of "SetUno!" when down to one card.

I've only played the game with two players, but I see no reason it wouldn't work with more (I'll probably try that soon and update here.)

And yes I like the name "Setuno" for this game.

I feel like Set cards could with not to much trouble be adapted to work like other card games as well.  Maybe I will try something with a bit more strategy than Uno next, like Rummy?