This is a threaded adapter to help you drain the last drops of liquid hand soap from a refill bottle into the dispenser. (It's not useful for the initial refills, but those shouldn't be a problem.)
I don't know how standardized the thread sizes are on these bottles, so here are the measurements for the ones I designed for:
- Top: Equate 56 oz. refill, 32mm outer diameter, 30mm root diameter.
- Bottom: Softsoap 7.5 or 11.25 oz., 27mm outer diameter, 25mm root diameter, 20mm inner diameter.
Both threads are 6 TPI (4.23mm pitch), which does seem to be fairly standard.