It sounds a lot like most US hot water systems, other than two things. One, the hot water heater isn't usually inside the house. It's almost always in a basement or crawlspace. Although I used to live in an old apartment that had one about the size (linear cabinetspace-wise, anyway) of a dishwasher.
Yes (central heating threads always seem to get as complex as food threads, without actually being as much fun), that's usually the size of the boiler (hot-water heater) in the UK too; the boiler usually isn't upstairs in the airing cupboard though, it's usually downstairs in the kitchen. Most are dishwasher-size, though my house has a very small kitchen and its boiler is half-dishwasher-size, which I hadn't previously seen. From the sound of it, do US ones tend to have the heater and the tank in the same unit?

And tangentially (I've long forgotten what this thread started off being about), is saying "other than two things" instead of "apart from" or "except for" standard US English? It looks very odd in UK English.

Two, the way you put it, it sounds like it's on a timer, which the ones in the US are not. They just detect the temperature of the water in the tank and heat if it's too cold. I'm not an expert, but I think it's probably about half-and-half gas units versus electrical units, with the electrical ones being immersion heaters.
The UK ones are thermostatic too, but even if the thermostat says fire it can be overruled by the timer telling it not to fire. I'd say they're usually used on timer mode except in the depths of winter, when you might switch into 24h mode.

It's all complicated by the fact that, usually, the boiler actually heats a separate "primary" hot-water circuit, which heats the "secondary" circuit for hot tap-water through a heat exchanger. The house's central-heating radiators are pumped directly from the primary circuit, and so, because of the way it's plumbed, you usually can't switch on the central heating without also getting the hot-water heating.

Peter