We just celebrated both our grandsons birthdays at the ASMo and I mentioned it when we made the reservations and when I made the PS for Crystal Palace for breakfast and the Rainforest Cafe for lunch.
When we arrived at the Magic Kingdom, I went to City Hall and got two birthday buttons. All day long, CM's and other guests wished them Happy Birthday, asked them how old they were, etc. At breakfast, the table was decorated with MM confetti and baloons and they had a birthday march with the characters around the restaurant and at the end of the meal, they brought them each a cupcake and a card signed by Winnie the Pooh, Tigger, Piglet and Eyeore.
When we got back to our room, there was some MM baloons attached to birthday cards with a picture of MM and signed by several characters.
At Animal Kingdom, I again got birthday buttons and the same thing happened. We had lunch at Rainforest Cafe and they brought them a Valcano and sang Happy Birthday to them. All in all, they had a great birthday celebration.
Like it was stated in other posts, make sure you mention it at the restaurants when you check in. We reminded them and they said to remind your server and the table was decorated when we got there.
At a couple of the fast food restaurants in the park, the CM's behind the counter gave them free dessert when they saw their buttons.
Unless you specifically order and pay for something yourself, that's about what you can expect. At least, that's what was done for the two boys. They are 5 and 7 and still talk about their birthday celebration at WDW.