How is a man choosing to identify as a woman not as offensive as white people “identifying” as black? Being black is not a feeling, and neither is being a woman.
You are conflating race and gender as social constructs with race and sex as biological classifications. You are also doing a little trick with the concepts of choice and identity. In other words, your question presupposes similarity where there isn’t any while also blurring the idea of what’s really a choice. Allow me to point out the fallacy in your framing with the following two statements:
A man who chooses to identify as a woman is merely in drag, whereas a male who has no choice but to identify as a woman is a transgender woman.
A white person who chooses to identify as a black person is merely in blackface (or co-opting blackness or appropriating black culture), whereas a person of European descent who has no choice but to identify as a person of African descent isn’t actually a thing.