Cool button. Just did some research on British Royal Navy buttons.
Here is a nice link. I don't think I saw yours there, but there were a lot of similar ones pictured. Navy buttons
I don't think your button is American military. Tice does not show one like it in his book, and in Albert's book most of the buttons that show just an anchor have the line (ain't no ropes on a boat) wrapped around the anchor shaft. There was an exception to that but it was a once piece button. Looking at the back, I agree it has some age. I would suggest a foreign navy perhaps.
Check out some of the French navy here. Not exact but getting close.