When signing up with websites I do not use my correct month of birth or if the choice is only in years, I offset my age by a year or two.  This is to prevent too much of my personal information being stored on insecure sites.
Most web sites are much less secure than banks, and even banks have a problem keeping your information safe. 

I know that many people lie about their age on dating websites but that's a diffrent issue. 
 
When you realize that the a website already has your email, your name and often your birth date they have half the information that is needed to build your identity and misuse your information. Not that it is there intention, but many websites fail to fully secure their customer data and often leak data like a sieve.

Personally I think it is good that people use their real age, in years but I advise them not to use their real day and month of birth. On the other hand maybe websites should stop asking for your full birth date. They relay don't need to know that.