There are only 902 American billionaires*, but this 'Redditor' claims to know nine of them. The wording implies these were chance friendships. It is a big answer, but not a great answer.
People tend to stick together. Once you know one odds become higher that you get introduced to a few more. Billionaires like parties as much as anyone else, so if you are not an insane stalker and get to know one well they will invite you to parties just to have a group of friends and now you are vetted so the others will not stay away as much. Different people have different sizes of friend circles so breaking in is hard.