It's an easy answer: Kirk is an anthropocentric bigot.
STVI
Captain James T. Kirk: Spock, you want to know something? Everybody's human.
Captain Spock: I find that remark... insulting.
He fosters this attitude among his crew.
Chekov: "We do believe that all planets have a sovereign claim to inalienable human rights."
Azetbur: "'Inalien'. If only you could hear yourselves. 'Human rights'. Why the very name is racist. The Federation is no more than a Homo-sapians-only club."
Chang: "Present company excepted, of course."
He pushes it on Spock. TOS: The Galileo Seven
Kirk: "You aren't going to admit that for the first time in your life, you committed a purely human, emotional act?"
Spock: [crosses arms and slowly shakes head] "No, sir."
Kirk: [laughing] "Mr. Spock, you are a stubborn man."
Spock: [raises eyebrows] "Yes, sir."
And Spock's used to this: STIV
SPOCK: There are other forms on intelligence on Earth, Doctor. Only human arrogance would assume the message must be meant for man.
KIRK: You're suggesting the transmission is meant for lifeform other than man?
Ignoring the fact that ST is created for a human audience, humans are consistently presented as the ideal life form and civilization. Many of the ethical crises we see in the series result from a Starfleet officer's inability to relate to an alien culture on it's own merits. Often times, Starfleet attempts to force Federation (read: Human) values on the societies it encouters, without regards for their own history, culture or structure. For instance: ST:VOY:"Critical Care", where the Doctor disrupts the social protocol of medical care because "valuable" citizens are receiving preferential treatment at the expense of the lower classes. A fantastic communitarian response, but that situation existed because of a shortage of resources and a crisis in logistics. The Doctor's interference will only exacerbate that problem.
Kirk does this a lot too. Almost always, he works to interfere, disrupt, and upset non-human societies by imposing his anthropocentric values on them. So when he says Spock is "The Most Human," it's a twisted way of respecting him by saying that Spock is everything Kirk thinks is valuable in the universe.