I presume running an electric shower includes heating up the water? Otherwise I don't understand how it could be so energy intensive. The energy is the mgh of the water lifted, plus the heat you put into it. Heat tends to be by far bigger in that calculation unless I've missed something.

Yes. They're likely referring to on-demand electric heating units that are installed near the shower head.

Our World in Data is based in the UK. Electric showers like this one https://www.mirashowers.co.uk/p/showers/shop-electric-shower... are more common there than they are in the US.