Here's the thing to remember about "ground". The word "ground" is commonly used, especially in the US, for two things: power "common", and earth ground. I'll attempt to carefully distinguish between them using these terms here, even though many people just say "ground".
Earth ground (often just referred to in Europe as simply "earth") is a safety thing. All the externally accessible metal bits of your cabinet should be tied together and connected to earth ground to prevent touch hazards between these metal chunks and, well, the earth. "Earth ground" is the 3rd prong on your IEC inlet. It's somewhat preferable to run dedicated, heavy gauge wires from each metal chunk back to a central point, but that's often impractical, and just daisy chaining them together is fine, too.
Power common, aka "the black wire", is something else. It's simply the return path for electric current from your DC supplies. All your signals and such are also referenced to this "common" line. This includes not just button signals but also your video signal as well as single-ended (non-balanced) audio lines like the line level audio outputs from your PC.
There's two common schools of thought on how to address this.
A common thing to do is connect these two "grounds" (earth ground and power common) together AT ONE POINT ONLY. No other connections are allowed as this results in "ground loops" which can cause audio hum, interference patterns in video, etc. Your PC will do this internally in its power supply. If any other connections exist, and there's even the SLIGHTEST bit of potential difference between them (which can happen for lots of common reasons), current will want to flow on your signal wires between all these connections to equalize things out. That's what causes the interference and creates the "ground loop".
The other option is to provide no path at all between the power common line and earth ground. This will work fine as long as absolutely no paths exist. However, the way switch mode supplies, like PC power supplies, are built, there's a small amount of current that wants to "find" earth ground, and it's surprisingly easy to create a "weak path" to earth such as through a person if they happen to touch both a connector and a chunk of metal. It'll also tingle (quite a bit) should you do this, but it's not particularly hazardous.
One thing to note is that arcade monitors will generally provide a tie between power common and earth ground. This is in a generally undesirable place and is difficult to avoid (since, for safety reasons, the monitor frame really should be grounded), but it's in a place that doesn't generally cause many problems. I have actually seen commercial arcade designs that don't ground the monitor frames for this reason, though.
The
Wikipedia article on "electrical grounding" will explain some of this in more detail. You can kinda think of the "power common" within your PC and/or cabinet as a "technical ground" as would exist in a TV/radio studio or a "signal ground" that might exist on a circuit board.