Sorry, try again.
Citing Shakespeare's culture when discussing Richard is like citing Tupac Shakur's when discussing Isaac Newton's. Shakespeare was born 365 years after Richard Coeur de Lion.
365 years after Richard I died that is.
From your quote, I think your problem is with the word "passionate". Richard and Philip are described as eating together (horrors), sleeping together (a common practice back then), being of one soul (is that gay?). The only other thing that might lead one to think they were gay was the word "passionate" which does not necessarily denote ardency of a sexual nature.
Besides all that, Henry II would hardly have marvelled to know that his son was homosexual. Rather, he would have marvelled that the 2 had a close friendship which would bode well for the expansion of his kingdom.