Such die and stock sets go back a very long time. Allegedly the Whitworth and BSP ones I ended up with via complex family inheritance route date back to before WW1. Something I have no way of proving but they are certainly early inter-war period at the latest.
My stocks hold all the dies in a set at once. Just release the clamp screw and ease the desired set apart to use. Nearly impossible to loose a die half as fiddling them out once the screw has been fully undone is tricky with about 10 thou clearance to work with.
Mine also have a 55° Vee roller and plain half "die" section to clean out threads. Not a universal fitting. Allegedly on gas fitters and the like sets.
Big problem with this form of die set is that there is no easy way to set to the correct size. The split sections are narrow enough to easily close up to well below the official core diameter.
I suspect the correct method is to close them up on a rod of core diameter to set the spacing.
Clive