It is a 3-piece "Staff" type of button, with extra-wide rim, which nearly always dates that type of button to being from the mid-1870s to the early 20th-Century.
About the various components of the logo on the button:
At the top, a spread-winged eagle stands beow a ribbon with a motto. (That's obvious, so let's go on to interpreting the other images on the button.) The man standing on the left does appear to be a sailor. There seem to be the top portions of "leaning flags" flanking each of the standing men. In the center, I think I see a crossed sword, rifle, and anchor. I seem to see another ribbon-with-motto at the bottom of the button. Some clearer, better-lit photos of that area would help.
The button shows images which are found on various State Seal buttons (such as, the standing sailor, the rising sun, and the standing eagle). I think the key to identifying it is in what is written on the ribbon at the top ...and also if there is a ribbon at the bottom.
Edit: I was busy going through the Albert button-book while Bramblefind was typing ...and she is correct. It is a Sons of [Union] Veterans, New York button, shown as VN-23 in that book.