I hope you don't mind but i rotated the first picture so it can be better seen. The button has good detail so I think it is more than a civilian one. And, yes, I think it is the face to a three piece button. The rim around it would have been attached to the back by a band, thus making it a three piece. It is not in Albert's button book that I could find. Hopefully someone will ID it for you.
It is the front of what is called a "false 3-piece" button... which means, a 2-piece button manufactured to closely resemble a 3-piece button. (See the diagram of button-forms, below). The rust-encrustation inside the back of your find indicates it had a tinplated-iron or steel back... which excludes it from being an actual Navy button. It is a civilian-usage "Fashion" button whose emblem is similar to a British Royal Marines button in service from 1833 to 1881, except it doesn't have the words "ROYAL MARINES" above the anchor. Your button seems to have been manufactured continuously from the mid-1900s to the present. Here are three photos of one for sale on Ebay. 1833-1881.