It's not really a false cognate. Menstruation does happen once a month...and you can call it your "monthly" or what have you. It's just very, very strange to hear someone use a word intending it to mean a monthly magazine and think of menstruation. Trust me, it is even stranger than it sounds in context.
Just think of all the contexts in which one might use "monthly" and how awkward it would be in those few seconds it takes you to tell yourself that it doesn't mean menstruation. Maybe you get together with friends for dinner once a month, maybe you have to pay your rent every month... trust me, it's not that often, but it's often enough to throw you off.
And it's only worse when you see things like this:
French word:
mensuel
English equivalent:
monthly
This is probably the least faux of my faux amis, but it's been seriously bothering me, people.