The imagery with the hat and axes harkens back to revolutionary times, I think. The 19 star theory is interesting and it could be the 30 count on the flags is an ideal at that time and not a reality.
I'm leaning toward the Daughters of the American Revolution. I'm told that for their National Conventions they adapt a design for a pin or piece of jewelry that members can proudly collect and display. That perhaps would explain the deviance in the star count?
I couldn't help but notice that the flags on this item have an incorrect number of horizontal stripes, assuming they tried to accurately portray the U.S.flag. That fact may make the number of stars in the field unreliable in terms of dating the piece.
I think the number of stars in this case was more a matter of what they could fit on there, and that it’s maybe early 1900’s.
The bit of blue enamel remaining might be a clue to finding another like it.