A coffee cup that is described as a 8 ounce (for example) will hold this amount filled to the top of the cup's lip or to some other arbitrary level?
I assume you're talking about understanding quoted manufacturer's dimensions of a drinking vessel (and not some measure of what an abstract "coffee cup" might be, as when people talk about a "4-cup" coffee maker or whatever).
To my knowledge, there is no standard at least in the U.S. for these sorts of measurements. My anecdotal experience is that cups and mugs are often slightly larger in capacity than advertised when filled to the brim, probably to take into account people's expectation that they can put a drink of X ounces in a cup made to hold X ounces without spilling it. However, that "error" seems to vary quite a bit. Years ago, I tried filling a few different mugs I had purchased just to get a sense of their accuracy in size. If they were supposed to hold X ounces, some would be full up to within about 1/16th of an inch of the top, while others had a gap of 1/2 inch or more below the lip. (Beyond my own anecdote, I also found more people's anecdotes here describing a similar trend.)
If I recall correctly, for whatever reason, there didn't seem to be as much error with glass measurements as there were with coffee mugs. Particularly if you see a very specific measurement advertised (e.g., "8.6 oz. capacity"), it seems more likely that the measurement is right up to the top. Obviously if you have a cup or glass with a line printed on it, it's generally more accurate. (This is more common in Europe.)
If you're asking about this to use a random cup as a "measuring cup," I wouldn't trust any listed specifications for capacity unless you actually measure the specific cup yourself.