Personalities are like diamonds, there are many facets to each. Best friends tend to have many common facets. Sometimes you see an unusual pairing of two friends, and it may be that they have only one facet in common but it is a big facet. Or their one similar facet was cut from the same experience.
I am good friends with some people who are good friends with others that I am not good friends with. My friends and I have similar facets, but they also have other facets in common with their friends that I don't share. It's not a bad thing, it just is.
Last night I was able to talk more in depth with someone who I thought didn't share many facets with me. We found out that we both have a big reading facet and so formed a friendship based on that. She is friends with people that I don't have any major facets in common with and so won't be friends (not enemies, but more acquaintances), but she and I found a big common facet that we can share.
Facets cut from the same experience can forge life long friendships between two people who have almost no other facets in common. This happens frequently in movies. Two or more people go through a harrowing experience together and come out friends for life, though they may not have much else in common.
Sometimes a small facet can become a temporarily significant facet. When people are in a new, unfamiliar and possibly uncomfortable situation (new job, new school, new neighborhood, new church, a conference or convention, in a foreign place), and they meet someone who was a mere acquaintance before, a small facet of common past background becomes a temporarily large facet to help establish familiarity until they both become comfortable in new surroundings and meet others that have more facets in common.
Meeting someone new can be a daunting process but looking for common facets will help establish a friendship quickly. There may be a surprising facet you didn't know about before with those you already know. People can get new facets as life progresses too. Remembering that people are multifaceted diamonds helps make life easier, funner, and full of interesting potential friends.