I agree with moscafj’s general answer: there is no pan-Italian “rule” like this in Italian cuisine. It’s common to mix the two in many Italian regions, and it’s certainly common in other world cuisines.
On the other hand, I wouldn’t dismiss this story out-of-hand or as some quirk of one crazy roommate.
I grew up near an old Italian neighbor, daughter of Italian immigrants. Her mother-in-law also lived with her, a direct Italian immigrant born around 1900 (immigrated sometime before mid-century). It was an absolute rule in their house that onion and garlic should never be combined in the same dish. And yes, they associated this with “the Old Country.”
Alas, I don’t know what region they were from, and they both died long ago. And I wouldn’t pass on this anecdote here except I did a few quick internet searches and discovered that a few people have asked similar things on various internet forums concerning Italian cuisine. And both Mark Bittman and Gino D’Acampo have apparently passed on tales that Italian cooks have told them the same thing. (I didn’t track down links to the original sources, but I have no reason to doubt these forum posts are simply making this stuff up.)
I’ve eaten food in Italy that clearly contained both. I know Italians who like cooking, and I’ve never heard them talk about this. On the other hand, I’ve heard this particular “rule” a couple times, and it seems only associated with Italian cuisine. I’m not really interested in trying to track down more information on it, but it sounds to me like it’s at least part of cooking “lore” for some Italians, perhaps from a particular region or something.
EDIT: Just for one Italian source which seems to reference the issue, see here. After a discussion of separate uses for onion and garlic, the question of whether to use them together comes up:
Il dubbio, a questo punto, viene: è possibile utilizzare aglio e
cipolla insieme? La questione divide da sempre gli appassionati di
Basically, this passage strongly implies that there are passionate cooks out there who argue about whether it’s even possible to combine the two. (The link goes on to argue that it is possible, but nevertheless it references this question as if it were a common dispute.) The opening paragraph also implies there are traditional fixed rules that dictate the specific occasions when they could be properly combined.