Domestic and feral cats are not automatically asocial (hostile to other cats) so within a colony of cats they may form close relationships and when cats become friends they show a range of friendly behaviors towards each other including touching noses, grooming each other, rubbing against each other, playing with each other and resting together.
But if we are discussing a different situation such as a neighborhood where cats are allowed to go outside and the cats are strangers to each other, how does one cat make friends with another? There obviously has to be some sort of initial contact and in the video below you will see that the black cat on the grass is trying to make friends with my cat, Charlie, who is in the flower bed snoozing.
It seems that if a cat wishes to make friends with a cat, who he or she does not know, then the initial step might be along the lines that you see in the video. The video shows that the cat who wishes to make friends positions himself within a reasonably close – but emotionally comfortable – vicinity of the target cat and waits, and he/she may also demonstrate submissive and/or friendly behaviour to encourage friendship. Note: the behavior of this cat is not the behavior of a female cat on heat so this is not pre-estrus behavior in my opinion (wrong? – please voice your opinion in a comment).
At that point the cat who is being targeted for friendship has a chance to respond by, I suppose, approaching the other cat and perhaps touching noses which is a friendly greeting between cats who know each other. In a colony no doubt the ranking of the individual cats within the colony plays a role in defining who can be friends with whom but this is not a colony situation. Also the gender of the cats and whether they are neutered is very significant.
As it happens, my cat was approached on several occasions (see the photo above taken in very dark conditions during an earlier encounter) and on the last occasions he made it known to the other cat that he did not want to be friends. This resulted in a mini-fight because when I went outside to see what had happened having heard the screaming I saw some soft fur on the ground and that fur must have belonged to the medium long-haired cat who had approached him for friendship. It seems my cat had had enough and attacked the other cat to force him away (How do cats choose companions?).
I don’t know if there will be more attempts to make friends and more rebuffs by my cat. It could well happen and no doubt I will hear about it by the screaming that emanates from the garden!
It would seem that the way a cat makes friends with another cat is very similar to the way humans make friends with humans. It requires an approach together with signals and signs which indicate that the approach is friendly which then allows the receiving cat to respond positively or otherwise.