As we move farther into an era of almost unlimited communication, there really isn't any way to peg down 'this is an Irish' or 'this is an American' or whatever song. I think that for the most part the label refers to specific idiosyncracies in the musical structure, or lyric conceits, that identify it most closely with the music from that country, or that period.

I'm not really the person to expound too much at length on this, but after having been subjected to a friend with a Masters in Music of the Renaissance (English), there are enough distinctions between the different cultural musical styles that it is still a valid distinction when talking about music. Remembering that a lot of these songs were considered the 'pop' music of their days, look at what dominates the 'pop' music of individual cultures today... there're still differences between what are considered the best 'American', 'English', 'German', etc., songs. A quickly diminishing difference, yes, and a lot of cultural overlap, but still enough to be noted.

M