It's ironic that a song about irony contains no actual examples of irony. But since that's ironic, is it actually an appropriately named song?

The song contains many instances of cosmic irony[0], as well as a few instances of other types of irony (e.g. "Well isn't this nice?").

[0] - https://en.wikipedia.org/wiki/Irony#Types_of_irony

Rain on your wedding day is not cosmic irony or any other kind. It's simply misfortune.