My MC's birthdays:
November 11th.
May 5th.
October 10th.
February 2nd.
Get it?
I DON'T GET IT BUT MY BIRTHDAY IS OCTOBER 10TH AND I THOUGHT YOU SHOULD KNOW IT'S A GREAT DAY TO BE BORN. THIS IS INFORMATION I AM GIVING YOU TO HELP AS RESEARCH OKAY GOOD.
Ahem.
Yeah, I generally know the month because that's important to keep track of things like whether they're older or younger amongst their classmates and whether they've been eligible for a learner's permit long enough to have their G1/G2/G yet, and how old they would have been if they discarded their virginity two summers ago, blah blah blah. If the date become significant in the text or in some random episode of the character's past/future that I secretly know even if it doesn't make it to the page, then I know the date. But generally, yeah, just the month.
Although I do like it when authors make specific dates significant in their texts and then readers latch onto that significance and recognise those dates when they come around IRL. Like January 10th in
Looking For Alaska.