catholique is used to refer to something that is “the norm”, what all people do usually.
“Ce gâteau ne me semble pas très catholique” means that you find the cake does not fit the standards for being called a cake: because it is weird looking, or tasting strange or whatever.
This meaning comes from the history of France to have a majority of Catholics since a long time. (I guess)
Catholique can, by association with the church by that name, often mean “moral” or “correct”, hence the common expression Ce n’est pas catholique! Thus the lyrics of this song probably mean that they are “immoral” (i.e., not chaste) lovers.