That's the old school backing stabilizer equivalent of a layer of fiberglass...with a butt joint repair, the gluing area is small, and the cloth (whether cotton or glass) helps spread forces across a greater area and unloads the butt joint somewhat. There isn't room (thickness) to use a dowel or biscuit between the pieces, so a backing cloth layer is a good idea.

To repair a cracked dash bow, I would lay a strip of doubled-up 6oz fiberglass tape all the way across, from one end to the other, simply because these dash bows need a little bit of twisting and persuasion to get out and replace, and having one crack a little on the way back in would be a shame.

The dash bow's internal layup is actually in two halves (the wood plies/layers below the top finish veneer meet near the middle at a butt joint) and running a strip of fiberglass or backing tape across that joint would be peace of mind for me. The factory hit the joint with a couple of staples across the seam, more to hold it in position while the glue set up. Applied right (just enough resin, not goopy/drowning the fibers), the glass tape is thin, not visible, and there"'s even a handy recess for it to sit in.