AA, like UA and many other airlines, uses TIMATIC to verify passengers comply with applicable regulations. Save for a rogue or untrained agent, I’ve never seen AA enforce requirements beyond what each country actually requires.
That said, airlines can be subject to (severe) financial penalties if they board a passenger who does not satisfy the requirements of the arrival country, so they have an incentive to be cautious. I don’t see anything that precludes an airline from imposing other requirements that are at least as stringent as the applicable regulations, if they wanted to do so.