Basically anything not a cat or dog is an “exotic”, and most of them require more out of the owner than said dogs and cats. Let’s face it, a lot of humans can’t be trusted to care for dogs and cats, either.
My family has parrots, which are probably one of the easier exotics to care for (if you get one of the better for pets species) but they’re still more work than a dog or cat, much more social than a dog, less attuned to human body language (even the ones they learn to talk still don’t seem to “get” humans as well as dogs), have picky diet requirements, are are easily killed by common household items (avocados, teflon cookware, some types of air fresheners, and other stuff).
They count as a recent “domestic” animal in some species because we started breeding about 150 years ago (before that, nearly all pet birds were captured from the wild, so for the person asking “why haven’t we domesticated anything recently?” there’s an answer for you - we’re working on parrots) but they still have all their wild bird instincts. They’re hugely social, making a friendly dog look like an antisocial hermit, and if not given sufficient attention they can go batshit crazy psychotic - isolated parrots not only have self-plucked themselves bare, but developed repetitive motion to the point of damage, and bitten their own toes off. A crazy parrot that’s crazy due to isolation is a very, very sad animal to see. They can’t always be rehabilitated.
We’ve been keeping parrots for about 20 years now (me - longer for my spouse) and we think we’re pretty good at it, but I’ve still got one bird with OCD tendencies because, I guess, two other birds plus hours a day from the humans just isn’t enough for him (he’s a feather-plucker - not bald, but definitely bald patches) and another bird with aggression tendencies we managed to direct to corn cobs and paper towel rolls instead of people (he’s a loverboy when he’s not upset about something). Birds aren’t mammals, they have different needs, very different ways of expressing themselves, and they’ve still got their weaponry. Large parrots such as macaws and cockatoos can be quite dangerous if mistreated and mishandled to the point they turn on their owners.
There are exotic pets that can be safely kept by the average person IF said person is willing to the required homework. There are others that are Just. Not. Good. Pets. In fact, most wild animals fall into the latter category, no matter how cute and cuddly they look.