Cats and dogs, certainly. Among the wealthy there was a vogue for expensive pedigree lap-dogs (pugs are mentioned in Jane Austen) but little girls also sometimes made friends with a large dog: a mastiff or Newfoundland.
Caged birds were popular. Not only the exotics (parrots, canaries...) but also native songbirds were kept - linnets, larks, finches, blackbirds.
Children of the lower classes where such animals were used for rabbitting and pest-control in barns sometimes had a pet ferret or polecat. Though less intelligent than cats, these can be surprisingly affectionate if gentled and humanised. Mice could be, and were tamed. Boys were more adventurous: the Duke of Wellington intervened to make sure a little boy's pet toad was properly looked after when the owner went away to school. We know that older children sometimes tamed corvids - jackdaws, magpies, carrion crows.
Hope this helps.