One thing I love about the mandolin is the wide variety of genres that it can be used for. I just got Don Julin's Mandolin for Dummies book and in it he goes over old time, ragtime, blues, bluegrass, Irish, Brazilian Choro, Italian, Classical, jazz, "Dawg" music, and more. The mandolin has found a home in each of these styles.
It's true that there are certain things you need to know about playing an instrument - in this case mandolin - regardless of what genre or style you want to play. Then, at a certain point if you want to become really specialized in one particular style, such as Irish, you have to really focus on techniques specific to that way of playing and purposely not let other elements seep into your chosen style.
While Irish tunes - the jigs, reels, slides, hornpipes, barn dances and so on - are among my favorite things to play, there's no way I could ever limit myself to just those Celtic tunes. There's too many other great numbers which fall under the broadly defined "old time" umbrella, as well as the melody lines from popular songs, that are also fun to play and learn.
So, I'm down with playing mandolin, as well as with playing Irish music. Those can the same thing or different things.