They are fine! Floricanes get old! Some of mine do not do this, but Fall Gold and others do, no worries! The ones that tend to be spineless seem to do this.
I have Allen black raspberry and the canes are purple, cool looking plant, super aggressive grower though. it wants to root all over! 6 times I pulled cane tips out of the ground last summer.

So it will be easy to tell which canes you need to remove once they fruit.

Thanks for the reassurance. I've been growing raspberries for years, but this is the first time I've noticed this. It's occurring among all 4 varieties I grow. I always assumed the bark of a raspberry floricane was analogous to that of a tree, and if the bark was removed across the whole surface of a cane that it would die. Hoping for the best.

Spur blight can cause peeling canes, old canes often peel. Spur blight will leave dark spores in the bark. Remove canes if spur blight and use a fungicide But peeling canes alone is not always a problem. Spent canes peel too.
Mine have always peeled, so I don't really pay attention to it.