The awards were first given out in September1953, the year the episode was set, suggesting that Rosskoff had only recently gotten his, in the inaugural awards. According to the Star Trek Encyclopedia (4th ed., vol. 1, p. 351), this was one of the reasons given by episode co-writer (and executive producer) Ira Steven Behr for setting the episode in this year. In the "real world", the Hugo for Best Novel of 1953 was won by Alfred Bester for the The Demolished Man.
The "prop" was a real award, provided by Hugo award winner Rick Sternbach. He won the award twice for "Best Professional Artist". (Encyclopedia (4th ed., vol. 1, p. 351); ;