The chimes aren't actually musical instruments, they're "fashion accessories". So anyone can equip them, bard or otherwise. I don't think there's an item type "musical instrument", but I could be wrong. What you really want to do is craft a fashion accessory, then customize it into an umbrella. However, the simple instruments are the only fashion accessories that I've ever seen anyone able to craft. Are there others? And the instruments can only be crafted by bards.
So through an odd chain of logic, only bards can make umbrellas. Which actually makes sense, considering a bard's fine fashion sense to begin with.