More simply put, it came to acquire the general definition, "the expression of one's meaning by using language that normally signifies the opposite, typically for humorous or emphatic effect".

It's like rain on your wedding day.
It's a free ride when you've already paid

Well, life has a funny way of sneaking up on you
When you think everything's okay and everything's going right
And life has a funny way of helping you out
When you think everything's gone wrong and everything blows up in your face

may it be that it's not the described situations that are ironic but life itself ? Life is using irony to tell Alanis something