It seems to me that there are two different (and equally important) dimensions to finding a play partner:
a) points/domains of commonality-- interests, goals, etc.
b) similar developmental readiness/needs.
Otherwise, this isn't actually a two-way 'friendship' at all. My DD had no actual "friends" until she was in her teens. The latter reason was a problem with peers who had commonalities with her on point a, and adults (who have a better understanding of the social give-and-take of authentic relationships) didn't really share her interests much of the time, or didn't take her seriously as a child.
I vividly recall playdates and opportunistic interactions with other children... my DD was always polite and reasonably gracious with agemates, but it definitely wasn't doing anything for HER. Other kids loved her, because she was the perfect friend, basically...
but she seldom sought interactions with children that she'd gotten to know. She was indifferent, mostly-- at least emotionally. Most of them (and their parents, for that matter) would have been shocked to learn that, however.
She was like a tolerant, benevolent older sister or favorite babysitter on some level. But it wore her out. Particularly the penchant for most kids under ten to be inflexible/incapable of perspective-taking enough to flex THEIR play preferences in any way. She got tired of playing the same thing. time. after. time. after. time... because it was boring.
She knew that those playmates couldn't go where she would rather have gone, in terms of play, so she didn't impose her preferences on them... she would just get tired of it and wander away after a bit rather than losing her temper when a playmate refused to respond to, "I'm kind of tired of playing _______ now."