Lots of dance tunes have been used as the basis for modern songs; not always with very impressive results. It's pretty unlikely that somebody would confuse St Anne's Reel with the Dublin Reel , unless they were very absent-minded. Both have "reel" in the title, of course.

The Canadian band 'The Irish Descendants' has recorded a song called Dublin Reel; you can hear a clip at http://www.irishdescendants.com/sofar.html. Another site lists lyrics, but indicates that the writer prefers that they are not displayed.

Somebody called Carl Corcoran has recorded something called Dancing the Dublin Reel. He appears to be a singer-songwriter, so it may well be a song.

Both those found via google, incidentally. They may even be the same song for all I know.