It's been pretty clear that the keeping of animals (exotic and otherwise) is something that many of our characters are interested, so I did a bit of looking into what that was like in the late 1700s and early 1800s. I found that it was apparently a lot more prolific than I expected, even commonplace. Even the common folk, especially in London, had easy access to pay to go see animals as exotic as tigers and lions, to the degree that the roars of big cats was noted to be simply one of the "sounds of the city." That said, given the changes in history in this game, I imagine that it's slightly different in this setting (although not entirely).
In the real world, owning exotic animals was very much an imperialist symbol -- in addition to simply being a sign of wealth, the English took a particular pleasure in displaying beasts from places they had conquered. Obviously, most of these animals were poorly understood and perhaps even more poorly kept, many surviving only a few years after being brought to England (seemingly with the exception of parrots, which were common pets for the upper class, and the kangaroos kept and very successfully bred initially by the queen). As if to highlight this fact, nearly all big cats were commonly called tigers (or tygers), and it seems as though just about any small rodent-like animal was called some kind of hare. Of course, interest in biological science was also rising at this time (perhaps because of this), and so animals were also valuable dead, as their pelts, bones, and other parts of their body could be more... intimately studied after death.
In this setting, however, I imagine that while this remains partially true, there would be some changes. Obviously, humans forever remain fascinated with animals, especially the exotic (and often the deadlier the better as well). However, since imperialism never really took off in this timeline, I imagine that the keeping of exotic animals is more of a curiosity, source of study, and symbol of trade wealth rather than that a symbol of conquest. Since the English maintain a better relationship with the cultures originating from the same places as these animals, they likely have a better understanding and greater respect for them, leading to better care and (at least slightly) more accurate identification. They still, of course, probably wouldn't commonly receive the type and level of care that animals do in (good) modern zoos, probably with less effort put into emulating natural habitats, and conservation of the species, but some effort would probably be put into providing, at the very least, a home that is not entirely unsuitable or deadly. It also means that exotic animals are probably slightly less common than they were in the real world, as better care means more expensive care, and more respect for their native lands means that rare or valuable animals are probably less likely to be stolen or poached. Instead, I imagine keeping exotic animals would be a sign of substantial wealth and/or strong overseas trade relations, having to buy the animals (or get permission for capture) from their native cultures, or be gifted them.
Of course, not all valuable animals are ones from overseas, and there are plenty of domestic or native animals in which their owners are quite invested, like purebred horses and dogs, or valuable livestock. I imagine much of this remains unchanged from the real world, but I would have to look into it a lot more to offer an opinion (if you're knowledgeable on any related topic, please feel free to contribute that knowledge!).
In any case, I thought this could be an space in which those with characters who are interested in either exotic animals or more domestic ones could talk about the types of creatures their characters own or are interested in, from tigers to hunting dogs. It could lend to a good subject for which characters to correspond in play -- inquiring about animals they'd like to get their hands on, breeding prospects for ones they already have, or just plain curiosity.