Doesn't that kinda negate the goal of using a fake birth date? After all, they don't really care whether you give your real birth date, it's enough that it is the same on that other website they get your personal information from to correlate.
Websites that use my real DOB are usually linked to my identity, so I’m much more concerned with protecting them from, eg social engineering attacks [0].
[0]: https://gizmodo.com/how-i-lost-my-50-000-twitter-username-15...