That's a rotating spindle, which indeed can catch a loose thread and reel it in. An oscillating cuter moves back and forth, so it unrolls just as much as it rolls, with no net pull.

And if the oscillation strokes are short enough it can saw rigid material while just vibrating jiggly flesh (this is how the saws used for cutting off casts work). Though cardboard is also pretty floppy, so the mechanism here is probably different (mostly the puck guard keeping fingers out)