Ad Duggap mentioned, there are many 20th-century imitations of the US Navy eagle-on-amchor button, manufactured for use on Civilian clothing, and therefore known as "Fashion" buttons.
Your button is definitely a Fashion button, not an actual US Navy button. The civilian Fashion imitations typically show the eagle with very round-topped wings, a very "puffed-out" breast, and the rope does not pass through the ring on the anchor's top. That's what's on your button. The imitation US Navy buttons can still be bought today at fabric-supply stores like Jo-Ann Fabrics & Crafts.