The factors which are need for something to be a very popular pet among all classes:
Not being dangerous, meaning animals like lions, hippos, bears and chimps are excluded.
Needing an affordable amount of food, meaning friendly animals like cows, horses and sheep are excluded
Not needing large amounts of space, meaning that like the animals above are excluded but also large fish that need ponds like catfish.
Not needing difficult to aquire food, meaning a lot of snakes who only eat rodents are excluded aswell as sloths who only eat one leaf.
Not needing unaffordable equipment like salt water fish need (as opposed to goldfish.)
Not being too unruly like squirrels or monkeys.
Not being too smelly like foxes (who's urine is extremely smelly and who intentionally pee where they live) are skunks (though skunks can have their glands removed.)
Considering this a few animals could have gained more popularity these being;
Rats and mice would could be just as popular as hamsters and are actually more friendly.
Insects that have reasonable diets could be more popular filling a similar niche to fish being placed in tanks.
Small tortoises could be popular in suitable environments (they deal badly with cold due to being cold blooded.)
Small Fish like tetras and guppies.
Chickens and ducks could work but chickens are easier to keep.
Pigeons are tame naturally and not the most expensive, doves are the same species and are similar but have a better reputation.
Ferrets could also be more popular and are similar to cats but more agressive.
For upper classes the situation is different, since they have more resources they can have bigger pets, like how horses are popular with the British upper class. The animals that could have been popular with them are:
Cheetahs, which are already quite popular in the middle east could have been a popular prestige animal, maybe if cheetah racing got popular with European elites they could be spread worldwide. They are also not as dangerous as people assume and have never killed a person, they are a lot smaller than other big cats.
Large and rare fish could become popular as a status symbol, maybe inspired by the koi keeping in east Asia but it could expanded to include any expensive fish or even crusatceans and cephlapods. Jelly fish are also very pretty and could easily be used decoratively but they are hard to maintain.
Capybaras are friendly but need large amounts of water and so could be popular although the barrier to entry is lower than others in this list.
Exotic birds like peacocks could be big but they are agressive.
Swans and geese could fit into this too, they are actually easier to tame than a lot of bigger birds since they have a history of being close to humans but they are already reletively popular.
Falconry and hawking used to be popular in Europe and could have stayed popular leading to birds or prey being popular pets.