From the other side here.
Female, one of my best friends is a gay male. I love him to bits, and am unusually demonstrative with him (more so than with my girl-friends). He doesn't like very feminine gay guys ("flaming homos") and neither do I... actually, we both don't really like extremely (stereotypically) feminine people - gay or not. So I doubt that I would ever have a gay buddy go "shopping" with me for clothes. But BFF and I are very into the same things - we have the same taste in movies, books, music, coffee etc. So we have a tonne of common ground.
He just... "gets" me. I think of him as being very much more open-minded than most guys, with no homophobia (also a plus) and a lot less stereotypically-minded. We hold discussions where we can agree to disagree, which is pretty often considering that we have very different political views. His feelings don't get hurt when I say that I disagree. He is awesome, and has a similar absurdist sense of humour. I'm not sure if the reason why we connect is so much that he is gay and I am female, more so than because he is himself and I am myself.
All I know is that sexual attraction is completely not a factor. I could (and would) hug him without worry of it being misconstrued. I feel completely secure in our friendship, and I know exactly where I stand - and he knows where he stands with me. I also know that I'll always be there for him, and vice versa.