I'm pretty certain this button relates to the Katy railroad--also known as the Missouri–Kansas–Texas Railway. Incorporated in 1870, it came to serve an extensive rail network in Texas, Oklahoma, Kansas, and Missouri. In 1989 the Katy was merged into Union Pacific Railroad system, and much of the Missouri track line has been adapted to become the Katy Trail bike path.
A great gift for the railroad aficionado, or for your favorite Kate, Katy, Katherine etc! 1"diameter and in great shape. Stamped on the back St Louis Button Company, St. Louis.