Is this a "Ruptured Duck" piece? Looks to be a cuff link. It has a round disk on the backside connected by a small diameter piece. My mom may have another one.
It looks to me to be the Honorable Service (aka Ruptured Duck) lapel pin, which was worn by veterans on the lapel of their civilian clothes.
This is one piece and the back is a flat disk. If you look at the picture, you can see the bottom disk has the same diameter as the front. Unless worn in a button hole, you can't pin it on.
A pic of the back would have been helpful. However, I found a thread on the usmilitariaforum.com which may address the original question. One of the members there apparently came across 3 pair of "Broken Duck" cuff links.
i did some searching and did find it mentioned as a lapel button, also. There seems to be quite a few of them.
https://www.etsy.com/listing/163637885/ww2-era-military-honorable-discharge?ga_order=most_relevant&ga_search_type=all&ga_view_type=gallery&ga_search_query=duck%20lapel%20pin&ref=sr_gallery_3 I think what you have is a lapel pin. At the end of WW2, many men's jackets had a lapel button hole (several of mine did as a kid). Rather than poke a hole in fabric many opted for a screwback pin that went through that button hole.