All the detail that is recessed (incuse) on the finished chip, sticks up above the surface of the metal dies. It's not hard to polish the die and achieve a mirror surface on all the chip's incuse features at the same time. The canes are not so obvious on most chips as they have a smaller area to reflect light.

Proof coins are made the same way by most mints.. the dies are periodically abrasive-blasted (to give a frosted appearance to the raised devices) and then polished to make the field a mirror. I don't know that this is the way C&J chips were made, but think it's likely the polishing, combined with finer milled clay, is what gives them their distinctive appearance.