Is it possible to redesign a portable electronic device such that it can be charged by a domestic microwave oven?
Sure. Place a \$\frac\lambda2\$ dipole in your microwave oven and see how the kV's spark over the feed point. You could of course harvest that power. It's an EM field – what did you expect?

So, the proof of concept being trivial, making a device that makes this safe and reliable to use is pretty much impossible – the microwave oven is designed to produce a standing wave (admittedly, that they typically try to change in shape with mechanical mode mixers) inside the oven cavity – you could as well hit one of the zeros in that standing wave and not receive much energy at all. 

Another problem is that what works with your intentional dipole works with the traces and metal parts of your mobile device, too. And since the wavelength of a microwave is in the same orders of magnitude as your average phone's dimensions, that is bad news. You'd have to essentially shield your phone from those in order to not fry it. And do that to a high degree of isolation, with a high degree of uncertainty in the frequency you block – microwave oven cyclotrons are truly not designed for highest frequency stability or accuracy. Sadly, shielding the device from microwave radiation strongly clashes with the purpose of your phone. You'll notice that e.g. WiFi in the 2.4 GHz band shares the frequencies with microwave oven. If you shield your phone against oven wavelengths, you'll shield it against WiFi, too. Your customers will be thrilled to have such a nice rechargeable useless thing.