It's not really a character thing as much as a logistics thing, in my view. Or maybe the characterization reflected the logistics of having a pet while traveling regularly.
For Highlander, they were all characters with a semi-nomadic existence who traveled between Paris and Seacouver twice a year, at least, and who made jet-setting trips across the world at a moment's notice like the Highlander characters did, pets were a non-starter and the fans would have noticed that they were constantly being abandoned for adventures. Even the humans moved around continually. Can you see a Watcher trying to take care of a pet and feeding/watering them regularly while tailing an Immortal?
Plus the quarantine requirements for getting a pet across country borders would be difficult - not impossible, given that virtually every lead character except Richie and Amanda seemed to have gobs of money built up and could pay their way through every obstacle. (And Amanda just stole Duncan's credit card or stole the money for that.) Logistically speaking, it was a lot to write around and have HL characters who were able to maintain their pets with actual empathy and feeling.
The Forever Knight characters who had pets were all human (to the best of my memory) and were rooted in one place, able to care for pets easily without needing to move them around and deal with immigration/emigration issues or matters of international law regarding animal importation/exportation on a regular basis. The regular vampire characters didn't have pets either - and if there were secondary vampires who had them (maybe someone had rats and didn't eat them? I don't remember), they might have been more rooted in place. Even the vampires were less nomadic than the Highlander characters - it's easier to hide from the daylight when you stick around in one corner of the world and build up your hidey-hole, only interacting with your own kind. But also for some vampires, there's maybe the urge to eat your pets.
OK, I actually have thoughts on this, but it's not what you think it is.
For Highlander, they were all characters with a semi-nomadic existence who traveled between Paris and Seacouver twice a year, at least, and who made jet-setting trips across the world at a moment's notice like the Highlander characters did, pets were a non-starter and the fans would have noticed that they were constantly being abandoned for adventures. Even the humans moved around continually. Can you see a Watcher trying to take care of a pet and feeding/watering them regularly while tailing an Immortal?
Plus the quarantine requirements for getting a pet across country borders would be difficult - not impossible, given that virtually every lead character except Richie and Amanda seemed to have gobs of money built up and could pay their way through every obstacle. (And Amanda just stole Duncan's credit card or stole the money for that.) Logistically speaking, it was a lot to write around and have HL characters who were able to maintain their pets with actual empathy and feeling.
The Forever Knight characters who had pets were all human (to the best of my memory) and were rooted in one place, able to care for pets easily without needing to move them around and deal with immigration/emigration issues or matters of international law regarding animal importation/exportation on a regular basis. The regular vampire characters didn't have pets either - and if there were secondary vampires who had them (maybe someone had rats and didn't eat them? I don't remember), they might have been more rooted in place. Even the vampires were less nomadic than the Highlander characters - it's easier to hide from the daylight when you stick around in one corner of the world and build up your hidey-hole, only interacting with your own kind. But also for some vampires, there's maybe the urge to eat your pets.