Browning died before even the High Power came out, and he had little to do with the final design. But FN had found that the Browning name had magic, so they continued to use it. The interesting thing is that Browning never really "worked for" FN. He was an independent designer who could sell his ideas to any company, depending on where he got the best deal. After his death, his family set up a company in the U.S. that licensed his designs.
But back before WWI, Colt and FN made a "no competition" agreement that Colt would not sell its products in Europe and Asia and FN would not sell in North and South America. This held until the 1950's, when those agreements were outlawed. Not until then was the FN "Baby" and other FN-Browning pistols imported into the U.S. by the Browning company.
Jim