The bag is also plastic..

It's a nylon (polyamide) mesh. It's not going to spew microfibers in the same way that a polyester fleece would. For reference, I found you some micrographs of the materials we're talking about:

poly fleece: https://www-tc.pbs.org/wgbh/nova/next/wp-content/uploads/201... nylon mesh: https://assets.fishersci.com/TFS-Assets/CCG/product-images/F...