I’m not sure how you get any sound at all with the power cord unplugged. 😁
But we generally hear ground loops as a hum, except!! Digital ground loops. They are a real thing that can occur with USB or coaxial digital connectors. I have never heard of this causing more than excess jitter though.
Seriously though, it should be 100% illegal to sell a DAC that doesn’t have complete galvanic isolation for it’s digital inputs.