Can a pizza cutter cut through fabric?
Unlikely.
There's a reason people keep separate fabric scissors from their regular scissors--cutting through fabric requires a very sharp blade, which you're unlikely to find on the average pizza cutter (designed to cut through soft cooked materials).
What you're seeing is a rotary cutter, a circular blade on a handle that's sharp enough to cut through fabric. Unlike a typical pizza cutter, rotary cutters come with blade guards and retractable blades to protect the user from cutting themselves. They're frequently used by quilters, who need to cut many pieces with straight lines, often using heavy plastic templates. In the images shown on the blog Quilter's Review's review of several cutters, you can see the safety features on the handheld rotary cutters. The CNC machine is using a similar blade, but as it's not being handled by a human directly, the need for the additional safety features is reduced.