The Biden campaign gave Andrea Lawful-Sanders 8 questions, of which she agreed to ask Biden 4 of the questions in a pre-recorded interview that was broadcast by WURD radio of Philadelphia on 04 July 2024. WURD has an online version (Archive link) where the relevant portion begins at 13:40. (Alternative Roll Call link to full interview since WURD deleted. Alternative youtube link)
Biden is speaking in a very choppy manner and listening to the whole interview it is understood that he isn't really meaning that he is a Black woman.
Biden says:
They should be in jail, and by the way I'm proud to be, as I said, the first vice president, the first Black woman [Lawful-Sanders says "mmhhm"] to serve with a Black president, I'm proud [to have a role in ?] the first Black woman [on ?] the Supreme Court
Listening to the whole interview, "as I said" refers back to Biden previously saying (around 8:30 in the online version) that he appointed the first Black vice-president and served under a Black president. Overall, people listening to the whole interview would understand that Biden is having trouble forming sentences at times, despite having known the questions in advance, but wouldn't consider him as referring to himself as Black or as a woman.
Update
WURD Radio has now canceled Andrea Lawful-Sanders and made the following statement about the interview:
On July 3, the first post-debate interview with President Joe Biden was arranged and negotiated independently by WURD Radio host Andrea Lawful-Sanders without knowledge, consultation or collaboration with WURD management. The interview featured pre-determined questions provided by the White House, which violates our practice of remaining an independent media outlet accountable to our listeners. As a result, Ms. Lawful-Sanders and WURD Radio have mutually agreed to part ways, effective immediately.
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Agreeing to a pre-determined set of questions jeopardizes that trust and is not a practice that WURD Radio engages in or endorses as a matter of practice or official policy.
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...mainstream media has historically ignored, marginalized and stereotyped Black people in their coverage. ... This practice of de-legitimizing Black voices continues today. WURD Radio is not a mouthpiece for the Biden or any other Administration.