I must be lucky, because I haven't really met many Black people with that mentality - maybe Caribbean people treat this differently? Most Black people I know know that they are mixed somewhere down the line, and they have no issue with anyone celebrating a mixed or multicultural heritage. I have almost always felt loved, accepted and embraced by the Black community, without having to deny anything about who I am or how my family is.
I don't see it as an either-or choice. You can be multiracial and be Black, and any other culture you want. I see myself as multiracial, Black and Latina, for example. I think if you WANT to call yourself white you can too, but in most cases white people and others won't accept this. I don't see why a person who is mixed with Polish and Black can't say they are Polish or Black Polish or whatever - if they have a Polish name, speak Polish, know Polish traditions etc. then they are Polish - since when do Poles have to be only white? I also think that IF more people classed as people of colour identified as white, it might take some of the privilege and exclusivity away from it. I also think in different settings, people can have different identities. In no way am I suggesting that anyone with some Black ancestry should be forced or guilted to ONLY call themself Black or not to acknowledge any other heritage. However, if they ONLY call themself biracial, I personally think they are just adding confusion and not unity. Just my opinion - I don't speak for all Black people.
Get used to me. Black, confident, cocky; my name, not yours; my religion, not yours; my goals, my own; get used to me. -Muhammad Ali