I'm not sure. I think it is just the concept that a things value doesn't change. It holds true in the equation you gave, but it doesn't really communicate the idea of simplifying math by changing things into easier math and then modifying to get the right answer. If that makes sense.
But that is the doubling trick. Right? Isn't the trick:
Which is the same as X+(X+1)=2X+1
It's not changing things to easier math to get the right answer. It's expressing the same equation in two different ways. First you have to recognize that 43=42+1, though. (Which, obviously, the OP's child did.)
I don't think this precludes learning, "when you see two two-digit numbers, add up the ones and then add up the tens column."