This is a feature of any device with a battery-backed real time clock (RTC). All x86 machines have this feature.
Pretty much anything with a clock that doesn't need resetting every time you turn it on can do this. An RTC chip is just a super low-power counter, and it costs almost nothing to toss in a wake-time comparator, so all of them do. When you set the wake-time alarm the host CPU really does power down 100% -- right after it sets the RTC chip's wake-time registers.
This is a long, long, long way from a microcontroller. It's not Turing-complete and has only a few dozen bits of storage for the counter. They're also manufactured on incredibly ancient fabrication processes (like 350nm until very recently) for lowest possible leakage.
The old BlackBerries allows you to "turn off" the device and have the alarm wake you up. It basically drops into a low power mode.
It was easy to make a phone dead. You yanked out the batteries.