A cat can be very particular about potty issues. Missing the box can be caused by a variety of health and/or behavioral factors. It's important to examine the cat's physical and emotional health, as well as traits of instinct to help figure out what's going on and find solutions.
Determining Why Cats Poop Outside the Litter Box
The H.I.S.S. Test stands for health, instinct, stress, and symptom solvers and can work well to narrow down what might be the problem.
Here's how it works:
Cats with health issues such as constipation can decide to "blame the litter box" and find other places to go. An older cat may also have problems with arthritis that makes it difficult to get in and out of the boxes, or to pose appropriately. Cats that refuse to use the litter box for one function but do use it for the other typically have a health issue prompting the problem.
Many cats prefer to have a different box for solids and another for liquids. Other times, they just don't want to share the facilities. This is why the 1+1 litter box rule -- one box per cat, plus one-- is generally recommended. So with two cats, ideally three litter boxes should be available.
Cats are by nature emotional creatures. It's wonderful if your cat gets along with his cat buddy, but there may be stress issues at work that are not obvious.
S=Symptom, Signs & Solutions
Cats recently adopted often take a few weeks or months to settle in and feel comfortable enough to reveal their personality and foibles.
It may be that your cat was still feeling a bit uncertain at first, and was willing to share the litter box. But after two months he decided to make a statement…on the rug.
Removing the rug and adding a litter box was a good move. Bathroom rugs for some reason seem to be common targets. Many of the bath mats have a plastic/rubberized backing that apparently smell like cat urine to the cat -- and so draw them to these areas to eliminate.
You cat should also be checked over by a vet. As mentioned under the HEALTH bullet, cats that use the litter box for urine or feces but refuse to use it for both, very often have a health problem causing this.
Second, while the two litter boxes are a wonderful solution, they need to be in two totally different places. Otherwise, one cat can "guard" and own both toilets and keep the other kitty away. Finally, take a look at the surface on which your cat prefers to defecate. Try duplicating that surface or add a THIRD litter box placed in another area of the house. For instance, if he likes tile, leave the box bare. If he's targeting paper, line the bottom with paper -- or if carpet, provide a carpet remnant. A very large box or one with lower sides may be more appealing if he's having difficulty maneuvering in the existing facilities.