I got the impression homosexuality in the Islamic world does have a long tradition
That's true, but it has nothing to do with Islam, theologically speaking. There will always be sexual and gender deviancy, in every society ever, regardless of its rules and scriptures.
I should add that I don't think it has anything to do with Greek culture. Arab culture has always been homosocial—and of course a lot of Arab culture has been exported to non-Arab societies through Islam—and it's no surprise that that can lend itself to physical and sexual intimacy among men. It still happens today.