I'm pretty sure that the person in Measure for Measure (Angelo) the original quote was said to, was not black, so the original use of the word ape was not as a racist insult. And I don't think AS intended it to be, or considered that it might be taken as such, in this scenario either.
It's a bit like when someone calls a child a 'little monkey' for being cheeky and larking about, fine if it's a white child, not ok if it's a black child, even though the intended meaning by the person saying it is the same.
People can say things with no intention of being offensive, but without full consideration of their audiences perception. Although, it would be difficult for everyone to manage this all of the time, surely it is expected that people may sometimes, unintentionally, offend, appologise and learn something.
I do think this incident will play into the hands of those who like to shout 'they're pulling the race card'.