I was brought up with Aspergers, so this is a personal reflection so no sources shall be stated. Also for those who claim I will be biased, I was bitten by a dog before I got a dog, and was petrified of animals, and found that having Sam helped me cope and learn to stop being afraid.
I had a pet dog, he died a couple of days ago. I had 3 other friends with aspergers and the only thing different between me and them was that I had a dog. (Parents in same income bracket, same parent relationships, families were all friends of each others). I was known to struggle extensively with change and socialising, instead of taking me to get labeled my parents thankfully addressed the actual problems! Sam was the solution to all my problems
When I was 9, we got a dog. I named him Sam. I actually feel that having Sam helped a lot, as when I felt unsafe to go outside I could take him with me, and felt it easier to relate to him than to other humans.
Unless there is a specific reason (allergy) it is actually considered that aspies (don't care if not approved term, it is what me and my aspie m8s use and find it better than a constant full label). If male are best to have a dog, and if female are best to have a cat. I know it may seem sexist but it is based on research papers I have been looking into after the diagnosis (explained below).
Whenever I got checked mentally, people thought I was just a very intelligent NT, until I had to move away for college where I didn't have Sam. Within a year I was diagnosed with mild paranoia and Aspergers Syndrome by a specialist.
Having a cat/dog gives an Aspie a "being" to communicate with who doesn't have the stresses of day-to-day life, making them consistent in reactions allowing us to unwind while not being withdrawn. Also it is well known that no cat or dog will ever betray you letting you avoid becoming too withdrawn.
Also for consideration: most aspies don't have imaginary friends as we can't make them... doesn't that mean that having a pet is just as important to our psychological development as having imaginary friends is to NT's?