Is it illegal for a minecraft server to ask for a DOB?

Only if they come by your house and force you at gunpoint to agree to enter it. They are ASKING you for it. You have a clear choice to enter it, or decide not to and not play there.

No, your Date of Birth is not protected information.

Illegal? No.

Sure, why not?
It is a private business.

No of course not. They can ask for anything they want. Then you can choose to play or not play.

Not only is it legal to ask for ANYTHING they want as a condition of granting you access to THEIR system, but if access to the system allows you access to age restricted material, it is actually REQUIRED to ask when you were born.

Most platforms have a minimum age requirement so asking for a DOB is pretty standard.

Of course not.