You know, for a teenager in boarding school, Kim spends a surprising amount of time with prostitutes. He watches Mahbub Ali go off with a "Flower of Delight" (1.191) in the first chapter (only for Mahbub Ali be drugged and searched, which isn't particularly romantic). The Amritzar courtesan on the train to Umballa is a prostitute, and Kim seems familiar enough with people like her that he can guess what she will do if he cries and pretends to be pathetic (the answer: give him money).
Kim visits a prostitute to buy brown dye to change the color of his skin when he wants to run away from St. Xavier's for the summer. And the blind woman Huneefa, who blesses Kim before he goes out on the road again with the lama, lives in a place called "The Birdcage," which certainly sounds like the name of a brothel. Still, in spite of the fact that Kim is deeply charming and knows how to make himself appealing to people, we don't really see Kim getting involved in any kind of romance over the course of his journey. Kim really keeps his mind on business rather than pleasure.
The one exception to this rule is the turquoise-wearing Woman of Shamlegh, who appears just as Kim gets away from the Russian agents with all of their baggage. In fact, the Woman of Shamlegh (her name is Lispeth, but the book almost never calls her that) is exceptional in a lot of ways. She offers both Kim and the lama a place to stay after their run-in with the Russian agents; she keeps the papers Kim stole from the two agents under lock and key until Kim takes them back again; and she finds out the whereabouts of the Babu and the two Europeans for Kim.
She is the ruler of her village (called Shamlegh, of course) in the Himalayas, and she has tons of authority over her people. In fact, she has husbands—plural—and she makes them carry the lama in a litter out of the village when he gets sick and can't really walk. The Woman of Shamlegh is clearly the boss in her part of the mountains.
Why does the Woman of Shamlegh do all of this? Because she is really, truly attracted to Kim. When Kim rejects her offer to come and live with her forever in her village, she is immediately angry. She wants Kim to understand that she is a mature woman with her own history: in fact, she once lived in a Christian settlement and fell in love with an Englishman. This man grew sick and then went away, never to return. The Woman of Shamlegh decided to go back to her own people and give up her Christian faith as a result. But she doesn't want Kim to think that she is desperate or unfamiliar with the world after Kim turns down the pass she makes.
Out of respect for the Woman of Shamlegh and the many favors that she has done him, Kim does something he almost never does: he actually breaks character. She thinks at first (of course) that he is a priest and a disciple of the lama's, since that is what Kim's clothes and speech tell her. But after she explains about her tragic earlier affair with an Englishman, Kim gives her a kiss and tells her in English, "Next time […] you must not be so sure of your heathen priests. Now I say good-bye" (14.145).
In the briefest of moments, Kim reveals that he is not what he seems to be. By showing the Woman of Shamlegh something of his identity underneath his appearance as a priest, Kim shows her that he trusts and respects her.
There was never any real chance that Kim was going to give up the Great Game—before he's even really started, mind you—to join the Woman of Shamlegh in her remote village. But he respects that she put herself out there emotionally. And he also likes that, "At least she did not treat [him] like a child" (14.148).
For the most part, Kim's coming-of-age is professional and emotional rather than sexual, but this little sidebar with the Woman of Shamlegh indicates that Kim's relationships with other people—and specifically, his relationships with women—may be about to change now that he has left school and grown up.