There is a phone number a cast member can call and when they give the phone to a little kid, there is Mickey or Goofy on a recorded wishing the child a happy birthday.
I totally forgot about the birthday greeting thing. We did that when we took a surprise birthday trip for the kids. IIRC, when setting up a birthday cake with Disney catering, they asked if we wanted a birthday call from Mickey. The message popped up into our room's voicemail a few minutes after we checked into our room. The kids were already in middle school, but it made them smile.