I put in some base cabinets last month. They made a weird humming sound whenever the refrigerator cycled. After some head scratching, I noticed a thin gap letting air vibrate a panel. I added a bit of foam tape to seal it. The customer and I laughed at the silly fix.
Parker, you're right that it's not true syncing, but it IS a kind of mechanical sympathy. That hum was the cabinet panel finding the EXACT resonant frequency the fridge motor put out. The foam tape changed the panel's tension, so it couldn't "sing along" anymore. It's a super common fix for those weird mystery buzzes.