We know that during the Empire children played with pets, toys and their friends.
There are mosaics and paintings that show children of Rome playing various games. There are even some writings that describe children at play. The writings of Horace talks about children riding hobbyhorses, building play houses and even making little carts and hitching mice up to them to pull them around.
Roman children played with balls and kites and made hoops with pieces of metal on them so that when they rolled they chimed like bells. Mostly though, boys played war type games. They practiced with wooden swords and played conquer Troy. Children would also make small figures and play with them. Girls played with dolls. Poorer children made rag dolls while the richer children had dolls of clay or wax some even had joints to bend and pose.
Dogs were a favorite pet of Romans but were not the only ones. Birds, including pigeons ducks and geese, were also pets. Late in the empire period, cats became common and might have been pets.