They are interchangeable. I guess you received a DOA product. I had already 2 P4P batteries which suddenly stopped working. You need to send an email to Support support.nl@dji.com, fill out a support request, wait for their approval and if they agree, you will get a replacement.
Not unheard of. When I ordered my first spare battery it was DOA from Amazon. Ended up returning it to them for a refund and went to Best Buy and they price matched it and it worked great.