Do the eggs really keep for up to a month?

Yes, they will be quite safe for that long. The marinade/sauce contains plenty of acidity, salt, and sugar- all of which will tend to make it nearly impossible for nasty things to grow. As a result of the brewing process, soy sauce is also full of lactic acid bacteria. Lactic acid bacilli also tend to crowd out all the other bacteria, though few things can survive in the salty environment anyway. Before salt was easy to obtain, the Chinese produced soy sauce as a way to stretch out their supplies and be able to preserve fish and vegetables in the days before rapid transportation and refrigeration. I made a big batch one year after Easter, when my kids participated in an egg hunt and came home with five dozen eggs to add to the four dozen we already had. They lasted, in the sauce mix and in the fridge for nearly a month before the last one was eaten. This recipe is very similar to mine, and I'd expect it would last just as well.