Common usage uses the name "electric shower" for the sort of thing you are talking about while a "power shower" is typically run from stored water using a booster pump for volume - particularly if the plumbing is gravity-fed as used to be ubiquitous in the UK.
The arguments against storage are that there are heat losses while it is standing and that it is more efficient to heat water directly, and immediately before use. This applies to "combi" boilers and "multipoint" water heaters as well as electric showers and over-basin tap heaters.
Losing the stored water certainly saves space in a typically tiny British house or flat, and where that house is occupied by one or two people who are out at work all day, what's the point having a cylinder of water gradually cooling down?
But where there are several people living together instantaneous systems struggle to cope - even a "big" combi would find it hard to supply enough heat for one person to take a shower while another was filling a bath and a third was doing the washing up, and most domestic electric supplies aren't suitable for more than one high power electric shower unit.
Stored systems can also integrate heat from multiple sources, so for our re-build we are intending to install a cylinder capable of being heated by solar panels, immersion heaters, a gas boiler and a log-burner's back boiler all at the same time if necessary. It will store the water at a much higher than normal temperature and we'll take heat out for both taps and radiators using heat exchangers. The heat exchanger for the taps (ignore the cylinder and scroll down to the Hot Water Module) is capable of being run in parallel, so if we find one is insufficient in the morning when three people try to take simultaneous showers we can simply (well, simply-ish) add another.