I know what the one on the right is, but needed some help on the octagonal one. Looks like a script "GR" for King George topped by a crown. A Rev War era navy button was also found nearby.
I could see that being something like a patriotic cufflink, but I have seen many different crown and GR buttons. I'm guessing they are some type of service buttons, maybe just not military issued. I've never seen a definitive ID for them. I do know during the Victoria era or maybe just a touch later post office used a 2 piece button with a crown over GR, so the early ones may be the same idea.