1) Of course, you can order a specialty cake.
2) But, eateries only do sporadic activities for special events.
3) There are just too many "special events" to make every event "special".
4) Most will wish a happy birthday and delivery a cupcake with a candle.
5) Beyond that, it all depends upon the INDIVIDUAL CM as to what happens.
6) Some examples of what can happen
. . . group wishes for a happy birthday (HDDR, CM eateries)
. . . character-signed cards or place mats
. . . 2-3 waiters singing happy birthday (but, seldom)
. . . cupcake with candle
. . . personalized menu with birthday greetings and person's name