Why does Kirk like Spock in TOS? Or Bashir like O'Brien in DS9? Why do you like your best friend?
My friends and I have fun together. Kirk and Spock have continually been at odds. Because of that there has to be a moment of respect.
My friends have qualities that I admire. They've shown me trust, respect, and loyalty. They have shown how kind they can be to me and to others. They have been generous and helped me through difficult times. They rely on me, I rely on them. They find me to be generous and a "good friend." We "take care of each other." That is a big part of our relationship. We BS in the morning over coffee and cigarettes. That's bonding time. If we need anything, the others would be there for us.
We have talked about why we are friends. We have done things and explained that it is because we are friends. As we have grown closer, we have talked more openly about where we stand. There's open communication. There's shared experiences. There is reciprocal love.
I can pinpoint when we became close friends. It was around the time that one of us went through back surgery and I was kind, asking every day if there was something I could do. We had time together, but that's really when things hardened into a friendship. Before that, we drifted into each other, got mad at one-another, and we remained acquaintances.
Open communication, trust, generosity, time spent together, love--all of that is at the basis for our relationship.
What this has to do with Kirk and Spock I'll never know, except to say that Kirk has very limited contact with Spock and that contact has been contentious. Comradery can build when you face death together, but that can't be the sole unstated reason in a movie that promises to be about relationships.
EDIT: In TOS, the respect of differences, led to a friendship (Spock and Kirk playing chess in their first scene together, "Not Chess, Mister Spock, Poker"). As they went through trials (like Spock almost killing Kirk in Amok Time), there friendship deepened. It took them 15 years to get to the point that these two do in 6 months (one of them dying and doing everything they can to save one-another).