It only matters on what console it plays on, it doesn't matter if it has Tekken or Capcom graphics on it.

When shopping be sure to look out what your buying it for, go to the 360 accessory section if you're shopping for 360 only, and vice versa for the PS3. Every arcade stick this generation(PS3, 360) are released for both consoles. 360 sticks also work on the PC

Arcade sticks are all about professional play, input is what matters the most, so go with wired, not wireless.

Hori is the leading brand, but also really expensive. If you want to start out small, I suggest you buy one of the cheaper Mad Catz brand ones.

I agree with GlockXIII.

It doesn't matter about the graphics, the stick will work with any game. They are just console specific; however, there are some websites that make custom arcade sticks that offer the option to work on multiple consoles (they run upwards of $300+ though). I've also seen some tutorials about how to mod the Mad Catz arcade stick to work on both the 360 and PS3, but does require some soldering.

In no way is Hori superior to MadCatz and I own two Hori sticks. The top end of both are excellent though.

Just don't get a stick with a MK button layout whatever you do. It just feels stupid and you will not be able to play any other fighting game on it decently.
I found it very easy to play MK on a stick with the standard Japanese layout though.

(Sticks also work great with any retro games you may own or download like the Mega Drive Collection for example)

All above.

But I'd say MK is a fine game without an arcade stick. MvC3 also works on D-Pad pretty well, but if you want to get really competitive in games like SSF4, you need to get an arcade stick. And I mean it.