It could indeed refer to rabbits' purported lusty sex drive, but here, more likely, 'breed like rabbits' (very superior old pale 44's #3) meaning 'reproduce in great numbers' (the obvious result of such behaviour). The Pope did indeed say that, but if he'd reflected first, he might have said something like 'have children responsibly', 'don't undertake the responsibilities of parenthood lightly'.
Rabbits' sexual behaviour, and the resulting overpopulation (ask any Australian) which results from it are cause and effect. Rabbits don't think before coupling; humans should. (Yeah, right, we always do...)
He was speaking in Italian, so we are discussing the translation of a translation for a start. For further context, he mentioned a woman about to have her 8th child in dangerous circumstances (to her health). He was definitely talking about having very large numbers of children, not the manner in which you conceive them. In French, doesn't 'go at it like rabbits' have a connotation of it all being over and done with quite quickly?