The object the Dodo gives as a prize in the Caucus Race is a thimble.
The Caucus Race follows Alice’s plunge into the pool of tears. The Dodo organizes a race in which everyone runs in an irregular circle, with no starting signal, fixed direction, or clear winner. After deciding that everybody has won, the creatures need prizes.
Alice distributes comfits from her pocket to the other participants, but she has nothing left for herself except a thimble. The Dodo ceremoniously presents that same thimble back to Alice as her prize. The scene comes directly from Chapter 3 of Alice’s Adventures in Wonderland, where Carroll uses the absurd ceremony to parody contests and formal political language.
The thimble is therefore Alice’s prize, not a prize awarded to the animals. That distinction can cause confusion, especially because the film compresses the book’s sequence and emphasizes the Dodo as the race’s master of ceremonies.