Since my earliest days on the internet and on the web in the early 1990's, I've always been nervous when services wanted by birthdate for (AFAICT) no good reason whatsoever. Since then, I've always used proxy dates as my "birthday" (dates that are easy for me to remember, but no one else knows) for services that don't legally require my real birthday (when filling out bank/govt forms, I will use my real DOB).
I'm thankful that I've done that, but of course it doesn't help when a FB friend helpfully sends me a "Happy Birthday" message on my real birthday or when my real birthday is available through so many public records. Sigh.
It is still not entirely clear to me why people allow the whole Facebook wall shenanigans. I've closed mine a long time ago, exactly because I never understood why would I want to mix in one place the public, the personal, and the intimate. The very idea of blurred boundaries is not sane, it's bound to keep creating problems (like the one you describe).
No, it's not that you could get in trouble for being under 13... it's the site that could be breaking the law by collecting data on children (without parental consent).
I'm thankful that I've done that, but of course it doesn't help when a FB friend helpfully sends me a "Happy Birthday" message on my real birthday or when my real birthday is available through so many public records. Sigh.