Wow, that brings back some memories. I sold a bushel or two of those things back in the 70's and 80's when I worked in a bike shop.

In the vernacular of the time it was called an "umbrella clip" due to its shape. An Italian company called Campagnolo originated the design but it was widely cloned. It was usually used to nestle the top end of a frame pump in the "umbrella".

Gershon is right, there was no matching clamp on the other end. The pump head itself was formed to rest directly against the frame, usually just atop the bottom bracket in between the seat tube and down tube. It was sometimes used on the top tube as well. Some Silca frame pumps were designed to rest directly against the frame on both ends and needed no clamp at all.