Candy's dog and the rabbits that Lennie dreams of tending are symbolic of the state in which these men live, representing the meager emotional world available to each man.
When Candy is introduced in the book, he is a jovial but friendless and rather defeated person. His only comfort and friend is his dog, a nearly lame and stinking old canine. We can see parallels between Candy and his dog.
Candy is the old, disabled ranch hand who is helpless to stop the shooting of his dog and who knows that he too will be banished when he is no longer useful.
The important fact of Candy's relationship to his dog, in the context of this question, concerns the isolation with which Candy is faced after his dog is killed. Though the dog was a burden to him, it was also his only attachment and comfort. Candy takes up a friendship with George and Lennie but there is no guarantee that this will last.
For Lennie, the rabbits represent a role reversal for him as well as a fulfillment of a childish desire to be near soft things. In Lennie's dream of tending rabbits on a ranch of his own, he will be the caretaker - in contrast to his current state with George.
We may see these attachments to animals as mere "creature comforts" craved by both men. However, in a world like theirs where friendships are often shallow and short-lived, their desire to own and care for animals represents a deeper lack of emotional security.