relish,
Your button find is likely from the following business listed in New York City, 1860:
Emanuel, Frederick
Tailor
113 Cedar
Source: Trow's New York City Directory 1861, compiled by H. Wilson, 1860
As fyrffytr1 mentioned, marked clothing buttons such as this were generally contracted out for manufacturing to an outside source, to display an advertising name of the merchant retailer or clothing manufacturer. In the New York Directory as listed, there are only a few shown with the surname Emanuel (last name). The given name Emanuel (first name) is much more common. The name is of Jewish descent. During the 19th Century, Jewish merchants and businessmen played a very prominent role in the clothing markets of America, as well as Europe.
In addition to Frederick Emanuel, the directory page below also lists several additional Emanual named businesses as well, in Trow's New York Directory:
CC Hunter