In big cities, specially in apartments in novogradnja, they may be pets, western style, with cat litter, visiting the vet, toys etc. In the suburbs and smaller cities, however, they are mostly not pets.
There, cats and dogs belong in the yard. Actually, cats have a free range and roam the neighborhood, roofs, attics and fences. The domestic dogs are generally on a chain, frequently hooked up to a long wire so they have the run from the gate to the backyard. Or there are stray dogs who still somehow get fed - by either sympathetic households or ravaging the garbage cans when they can.
Then there is poultry. Traditionally, there would be a whole section of the yard given to them, and a chicken coop or a shed with a roost. They'd eat whatever grows there (usually nothing if there are too many of them), kitchen garbage and leftovers, or would be fed regularly by premix fodder, which can be bought in specialized shops. These shops usually have all the garden stuff plus all the food for pets and poultry, and can be found near the suburbs. People usually keep chicken for the eggs, some even commercially (you can find them in near a supermarket, with a stool and a few cartons of eggs). They also keep them for meat, and any well seasoned housewife there knows how to kill a hen, clean it up and pluck it. The head and feet usually go to the cats and dogs.
Some even have few pigs or goats, although the space usually doesn't permit more than 2-3. Some know how to slaughter them, some call a butcher - anyway, it's traditionally done on the spot. Cutting it up, making sausage and other suhomesnati stuff is a trade known by many, and even such amateurs generally know how to do it right. In late november and early december, there's a common sound of the gas flamethrower (caled brener, from german Brenner - burner) which is used for scorching the bristle off the skin. Any decent household in the suburbs has the equipment - this burner, a kettle to boil the lungs, liver, head etc for some kinds of sausage, the sausage filling cylinder/piston, a meat grinder (manual, the electric ones being either unreliable toys or too expensive). Some even build a smoke chamber to cure the ham, sausages and few select cuts. Everyone prefers homemade.