This was either used as a wall soap dispenser (liquid or powder soap), or an automated drip dispenser to sanitize toilets.
The Rochester Germicide Company is still in business today after 132 years. They are currently business as the Rochester Midland Corporation. When they started in 1888, the supplied equipment and disinfecting fluid for restrooms.
This bottle is near perfect and very clean. It's life as a soap bottle has kept it clean, I'd say. This would make a great lamp if repurposed. The letters say Rochester Germicide Co NY Pat Jan 25 1888. 8.5 inches tall.