I've never become friends with any of the dozens of people with whom I've played chess in person (see below for why the qualifier "in person" is relevant) excepting one high-school classmate. A chess player is pretty much forced to suppress any natural human cooperative instincts like reciprocal altruism, instincts that are probably very important in the establishment of friendships. Also, sharing small pleasures seems important in starting friendships, and in chess the pleasure of one party coincides with pain in the other party.

Also, since the early 1990s anyone logging on to the Free Internet Chess Server can with an expected wait of less than a minute be matched up with another chess player of whatever skill level (more precisely, Elo rating, which is calculated automatically by the server) one desires. There is no need to remember the identity of who one has previously played against (although doing so will tend to increase one's rating a little since individual players have styles that can be learned and exploited in the game).