It really depends. DH and I went down for our 10th anniversary earlier this year, DH planned the trip as a surprise for me, and he made all of the ADRs.
At the Coral Reef, we got personalized menus and a glass of champagne each. At the California Grill, we got a chocolate "Happy Anniversary" sign on our dessert plates and they took our picture and put it in a card. At Boma, we got a card signed by the hostesses in their native languages (really cool!) and a plate of samples of different desserts with "Happy Anniversary" written in chocolate. At Teppanyaki, we got a piece of cake with icecream to share (in addition to the desserts we bought).
We didn't ask for anything at all; DH did have our anniversary noted on the ADRs. Each place seemed to know it when we checked in.