Sorry, but it is absolutely definitely a civilian Fashion Clothing Industry imitation of an actual US Navy button. The true Military-issue ones always had excellent die-work, meaning the details of the emblem (such as, the feathers on the eagle's wings) are sharp and clear. Also, the eagle on Manwayvan's button has very "round-topped" wings. His button does not appear in any book on Military Uniform buttons, such as the Albert book and the Tice book.
Here's a photo of an actual US Navy button. Notice how "sharp" the detail-work on the button is. Compare it with the other photo, which shows a cheap-quality, crude civilian-usage imitation of it.
Manwayan's button is indeed a very crude, cheaply made civilian-usage one, with very weak details, because the manufacturer did not want to pay the high cost of having a Master Craftsman create a sharply-detailed stamping die for making the button. It is most probably from a yachting jacket or an imitation navy pea-coat.