Sounds like she believes these control a multiwire branch circuit. What you have is 4 circuits in the space of two full size breakers, not a multiwire branch circuit.It definitely is not about the house panel. They were ok with that.
They were only concerned with this panel. She said she was concerned if one tripped, there might still be current so she wanted them to trip in tandem. You are right. I may have to call for more clarification.