Chef Mickeys does a celebration and normally bring out a birthday cupcake.
At Chef Mickey's birthday kids get a yellow napkin when non-birthday guests get a blue one.
This helps characters and Cast Members identify who's the Very Important Person
But, OP, just as Elise said, do not expect anything at restaurants. it's hit or miss.
DS almost always got a card, signed by characters, and a birthday cupcake with a candle, at character meals. Sometimes he even had a happy birthday song, but not all of the time. There was one meal where the Waiter made an "announcement" in order to draw attention so that other guests joined in for the happy birthday song, but, again, it's really hit or miss.
When you book restaurants, always make sure that you mention the birthday celebration. It doesn't matter if it's not on the day.
We went for 21 nights free
DDP last year, and we celebrated DS's birthday 21 times

Sometimes he received a free cupcake, sometimes he didn't receive anything.
If you're planning a "celebration", it would be safer to order a cake or something like that. Don't ask me how, I don't have a clue, I just overheard some guests talking about birthday cakes (or was it on the DIS ? ... )