I know that, when wearing a suit, a pocket square is usually a nice addon - but those are usually nothing more than ornamental, and men today rarely (if ever) carry a handkerchief for utility. When and why did the practice die out? :) [ Mostly referring to the United States here, I'm not completely acquainted with the rest of the modern world :) ]
The handkerchief was blown away by the Kleenex tissue. The 1930s were a really interesting decade for the stylish pieces of cloth. On the one hand, their affordability made them prime fashion statements for upper and middle class Americans when outfits were a bit further out of reach. Swapping out the color and design of your handkerchief really showed who was styling. A dash of color popping out of a pocket, purse, or belt line. Men took the style of the OP mentioned pocket square for their suits.
At the same time, the utility of the handkerchief was going down. Kleenex was introduced in 1928 as a disposable tissue. In 1932, the came up with a campaign slogan, "The Handkerchief You Can Throw Away!" and emphasized the germiness of reusable cloth handkerchiefs. Americans responded by purchasing Kleenex and using their kerchiefs primarily for decoration.