The PS4 doesn't run on a battery though, so if they can get roughly the same power with a much lower power consumption it's already really good.

That's a good point, hadn't thought of the battery.

Apple is claiming that the iPhone 7 has console-level graphics capabilities. If that's true (and I have no idea what that actually means), then Nintendo should be able to come up with something even better since they have so much more room for battery and have lots of expertise in making consoles.

It is purely marketing.