AMHA will put the chip # on your registration for you, yes. AMHR, I don't know.
Hunt House Farms used to microchip their horses and naturally had readers. They got the chips from AMHA & I suppose they still have them for sale....???? But, the main reason for them was to be able to identify with CERTAINTY, their own animals at the farm before administering vaccines, etc. Plus, as it was told to me, they wanted to be sure the farm help were telling them about the correct animal if they called with an "emergency" if they were away from the farm, they could be certain which animal was breeding whom, etc. Hey, a bay is a bay -- is a bay!
And a hired hand is a hired hand is a --- get the picture?
Most all "systems" have their faults. In our State, they are looking to identify the stock that is in the food chain, most importantly. When we raised Boers, they used tattoo and/or microchip for all registrations. In fact, I still have a few of the unused chips right now.