They are under the same company, but different stores although you can use either the Bergdorf or NM CC at both stores. I've purchased things from Bergdorf's but have not been able to return at Neiman's. I've had to send it back or wait until my next NYC visit to return things.
It's exactly like buying something from Gap company - you can use their CC at the GAP, BR, Old Navy, etc., but clearly, you can't return something from the Gap to BR, vice versa.