I think it's easy to see why anyone might want to have Colonel Brandon as a friend. In Ch. 10, Elinor's description of him (in defense against Marianne and Willoughby's unfair and unkind remarks) describes the sort of person I think anyone would like to have as a friend:

"He has seen a great deal of the world; has been abroad; has read, and has a thinking mind. I have found him capable of giving me much information on various subjects, and he has always answered my inquiries with the readiness of good-breeding and good nature.

... I can only pronounce him to be a sensible man, well-bred, well-informed, of gentle address, and, I believe, possessing an amiable heart."

Brandon is gentlemanyly, polite, kind to people, intelligent and can talk about many different subjects. It's not surprising to me that Sir John would value his companionship and friendship.

It's perhaps a bit more surprising that Colonel Brandon likes Sir John so much as a friend, for although Sir John is kind and jovial, he can also be unintentionally thoughtless and tactless at times. However, it is never mean-spirited.

Also, we are seeing Colonel Brandon as grave and reserved, but Elinor also observed this of him: "His manners, though serious, were mild; and his reserve appeared rather the result of some oppression of spirits, than of any natural gloominess of temper. Sir John had dropped hints of past injuries and disappointments, which justified her belief of his being an unfortunate man, and she regarded him with respect and compassion."

To me that suggests that he is or has been at other times more lively and less serious than he is at present in the book. If he is currently under some 'oppression of spirits' being around Sir John's good humour might be exactly what Brandon needs.