Yeah. In the US basically every shower is hooked up to the same hot water source as everything else in the house, which, traditionally, is a thirty to fifty gallon insulated tank constantly kept at temperature by an electric or gas heating device, called a "hot water heater", though there have been significant inroads made by tankless water heaters in the last ten years or so.

I'm sure that one of the reasons that we don't have electric showers is because the vast majority of power in a US home is 110V, which probably isn't even close to enough to heat water that quickly. (Generally, the only 220V connection is for a clothes dryer.)