The key characteristics of a good PM I'd identify are intelligence and memory.
The best PM I ever had the pleasure of working with couldn't code worth a damn. And yet, he could recite details he'd heard from developers 3 weeks ago, and apply the understanding he'd gleaned to current decisions. Furthermore, he had an unvarying nose for bullshit and ability to gently push if developers were trying to make lazy decisions.
Unfortunately, I don't think that's something you can test for or train. Either someone is that type of person, or not.
The best PM I ever had the pleasure of working with couldn't code worth a damn. And yet, he could recite details he'd heard from developers 3 weeks ago, and apply the understanding he'd gleaned to current decisions. Furthermore, he had an unvarying nose for bullshit and ability to gently push if developers were trying to make lazy decisions.
Unfortunately, I don't think that's something you can test for or train. Either someone is that type of person, or not.