He is both, and saying "BIracial" covers both. It doesn't automatically cancel one or the other. I can concede that it could be rude in the context of the conversation, possibly, but not that it is offensive to the person who is actually biracial, unless one is using it to negate the experience of being one or the other.
On further thought, I can agree that it would be rude/incorrect if one were using "biracial" to say "He's not African American, or he's not black." That's an extension of my last sentence - I was previously thinking more along the lines of someone calling him AA/black in a derogatory way (and that Ave would be correcting the wrong), but if they are being inclusive or having a specific conversation, I can concede your point.