It depends a great deal on whether you want to know if a particular pet is in a room, or if you just want to know that one of the pets is in the room.
If you want to know that one of the pets is in the room, but you don’t care which one, you can use any of the various “people counter” technologies.
If you want to know if a particular pet is in the room, then each pet is going to have to wear some kind of ID or you’re going to have to be certain that they were close enough to a microchip reader.
All of these concepts fall under the category of “microlocation.” It’s definitely possible, but the problem is it becomes quite expensive. Most of the devices use Radio frequencies which can pass through walls, which makes it hard to define a “room” the way people think about it. So you generally have to have at least three devices in each room, maybe more, or otherwise you will think that the pet is in the family room when in fact they are in the kitchen next-door or even the bedroom one flight above.
There are a number of discussions in the forums about various microlocation projects, but very few people have ever implemented them again because of the cost. It’s likely you would be adding around $200 in devices for each room, maybe more.
But check the quickbrowse lists in the community – created wiki in the project report section and look at the list for “presence and microlocation.”
http://thingsthataresmart.wiki/index.php?title=How_to_Quick_Browse_the_Community-Created_SmartApps_Forum_Section#Quick_Browse_Links_for_Project_Reports.2FQuestions
As a practical matter, many people with pets just rely mostly on interior cameras to see if a pet is in a particular room. You don’t have to add any additional devices, the total cost can actually be less than trying to set up a full microlocation system, and you don’t have to have the pet wear a device. So just another approach to consider. 
And more about microchip readers:
there are automated pet doors which have microchip readers built in, so if you were willing to put up a baby gate on each room doorway with one of these pet gates built into it, you could certainly track all the comings and going’s if all the animals are about the same size.
Where it gets tricky is if the dogs are much larger than the cats but the cats may uses the dog door. Because the cats probably won’t go through in a way that causes their microchips to be read.
Anyway, this is an option for pets that you don’t have for people but it does add a lot of expense to each room ( at least $300 per room) as well as looking a little weird. Or of course you could keep the regular doors closed and add the pet door in the wall or in that door. Again with the idea of forcing each pet to pass by the microchip reader as they go in and out of the room.
So doable, but not subtle.
