Since nobody else has replied:
There has been a lot of debate about whether those thin stamped sheetbrass anchor-with-rope-border 2-hole buttons are actual Navy buttons or civilian-usage "nautical motif" buttons. No such button appears in the Alphaeus Albert book on American Military buttons. The closest is a US Navy button made of plastic, with four holes (instead of two), and no rope-border. So, lacking any documentation that it is a US Military button, I lean to it being a civilian-usage one, or perhaps from some other country's navy.