I note that the third sense of the five choices offered in the OED seems the closest relative, so I wonder if the button-like protuberance was named after the frog fastening on a coat?

[Of obscure origin; perh. ad. Pg. froco (repr. L. floccus flock n.), which has much the same sense.]
1. An attachment to the waist-belt in which a sword or bayonet or hatchet may be carried.

2. An ornamental fastening for the front of a military coat or cloak, consisting of a spindle-shaped button, covered with silk or other material, which passes through a loop on the opposite side of the garment.

3. Comb., as frog-belt, -button.

© OED