Google’s “define” feature describes a friend as being “a person whom one knows and with whom one has a bond of mutual affection, typically exclusive of sexual or family relations.” The word “person” obviously rules out cats, but I think the rest of the definition holds true, save for the fact that, like other close buddies, they are often considered as being family members.
Connections with cats can be just as rewarding as those with humans, although the experience is different. With people pals, we rely so much upon either written or spoken language. Steven Hales, author of the book What Philosophy Can Tell You About Your Cat, points out that cats may have better access to our emotional ups and downs than do many of our human companions. Why? Their heightened sensitivity to touch, noises and visual signals allows them to understand your feelings even before you open your mouth to express them in some way. “Our cat companions, then, may be able to penetrate to the heart of our emotional lives in ways that our human companions cannot,” says Hales.
Therefore, I think that cats have friends -- including human ones -- but that these friendships are not precisely on par with those shared between people. In terms of what cats think, we cannot be certain, but cats have an incredible memory for good things, like warm laps and tasty food. However, a recent study in the journal Behavioural Processes determined that cats are drawn to people for reasons other than just food. The cat version of friendship is surely one of them.