Compare the equivalent product, rice. Rice is stewed grass seeds. You don't think it's a plant product? Are potatoes a plant product?

Grain alcohol, depending on how you prepare it, might easily retain no trace of biological material other than sugar and alcohol. That is not true of bread; bread is not heavily processed at all. You could trivially analyze any sample of bread and determine what plants it was made from.

Claiming that bread is not a plant product makes all the sense of claiming that cheese is not a milk product.