The Stirrup Shirt Stay can be seen as the third generation of keeping your shirt tight to your body.
Someone probably had a very bad experience with a straight shirt stay coming undone and probably received a shot to the groin from a renegade clip; with this new, painful experience, this someone probably had the idea to get rid of the clips on the socks and introduce a more reliable design. As the name implies, the stirrup shirt stay no longer has the sock clips.
Instead, the bottom portion of the shirt stay is worn around the foot like a stirrup of a saddle. It is certainly an improvement and reduces the risk of injury to the groin, but quality of construction and material are not much better than the straight and Y-clip shirt stay.