He's not ashamed, it's just as you said at the bottom: he doesn't want the prefix attached as if being a non-white hero makes him something less than bona fide and more akin to a dog n pony show wherever he goes.How is it c00ning??
....Then again, reading that scene once again, he does seem a bit weary that they are calling him Black. Like he's ashamed of his blackness.
But at the same time, he wants to be known about his accomplishments and his results, not his race. He wants to put that first over being known as some type of affirmative-action superhero.
Add to it that he's mixed and he feels being called "black spider-man" isn't technically accurate anyway. But it's not like being called the "black puerto rican spider-man" everywhere he goes would make it better.
"Oh shyt, LOOK! it's black puerto-rican spiderman!! "
"Damn, look how