In the book, Fudge is Peter's toddler brother. He's mischievous and often gets into trouble at home.
In Chapter 10, we discover Fudge has swallowed Peter's pet turtle, Dribble. Frantic with fear and worry, the boys' mother takes Fudge to the hospital. At the hospital, Fudge is given castor oil, milk of magnesia, and prune juice. The idea is to ensure Fudge will pass Peter's turtle out in his stools. In the meantime, Peter worries about Dribble; he's pretty angry Fudge once again managed to make life difficult for him.
In the end, Dribble is passed out in Fudge's stools; however, the turtle is dead, and Peter is upset about the whole affair. There is a silver lining to the whole mishap, however, when Peter gets a surprise from his parents. Because Peter has been a relatively good sport about the loss of his turtle, his parents buy him his own dog. Since Fudge is expected to try to appropriate (seize or take) Peter's new pet for his own, Peter's parents joke they chose a pet for Peter which can't be easily swallowed. That way, Peter won't have to worry about Fudge taking away his new pet.
So, yes, if given the opportunity, Fudge will likely take Peter's new pet for his own. Also, since he's only a baby, he will likely physically treat the dog less carefully. We already see evidence for this when Fudge first spies Peter's new pet. He immediately walks over to the dog and grabs its tail. Fortunately, Peter's father manages to free the dog from Fudge's grasp and tells the toddler in no uncertain terms that the dog belongs to Peter.