So, here's what I know. The bottle was produced by the Pierce Glass Company. They only operated one factory at a time, the last one being in Port Allegany, Pennsylvania from 1917 to 1987, when they were bought out by another company and ceased use of their trademark "P inside a circle". Since they only operated out of one factory, the 3 is not intended to identify its origin.
Here's what I think. The shape of the bottle, and the fluting around the bottom, makes me think condiment, not medicine or liquor. The small size rules out major sauces like ketchup and mustard, so I'm leaning toward hot sauce or some kind of other strong sauce. Pierce would also make custom bottles on request, so it's possible that you'll never find its exact original use, but not certain.