If you could blow your nose in it, it's a handkerchief.
A handkerchief can be used as a pocket square (before you blow your nose in it, or, if after, with the grotty bits unseen)
I think well-dressed men used to carry both -- one in the breast pocket, another (if the first was silk, for example) in a pants pocket.
The idea of transporting one's own snot after expelling it is abhorrent to other peoples, for ex the Japanese, who use disposable tissues. Now most Americans do as well.