A certain care is necessary with B16 chuck mount tapers. They come in two varieties, short and long, and are very close dimensionally to the Jacobs 33 taper. So much so that, depending on tolerances, it is possible to get wrong pairs to apparently assemble. Maybe even well enough to drill a few holes. This is most likely with no name or "sounds like a brand" bargain components.
I once had the dubious pleasure of sorting out a chuck and arbour pair where even the makers seemed confused as the measurements on both fell between all three standards. As the taper sides were barrelled rather than straight I eventually concluded that the (unknown) makers had concocted a sort of universal fit which would mate adequately with any provided sufficient pressure or large enough hammer were applied. Alignment afterwards being, most likely, somewhat imperfect.
Clive