Are the water year/hydrological year different around the world?
The water year does vary depending on the region, for example in Australia the water year begins on July 1. I am not sure about other regions, I believe the October 1st water year is quite standard across the northern hemisphere although may shift as you go far north to polar and more arid climates. I would recommend searching for similar links to the one for Australia to find the government defined water year, if it exists in your region of interest.

The purpose of the water year is to use, in lamens terms, the least exciting hydrologic time of year to avoid missing large events or fluxes based on the start time of simulation or data consideration, so depending on the annual hydrologic patterns in the given region shifting the water year start generally makes sense. For this reason as well, the calendar year is generally a bad idea for hydrologic models or data considerations, since you are then starting during a time of accumulated snowpack and can have mid-winter melts around that time (at least in the northern hemisphere, generally speaking).