It's more general than that:

If you use a card that earns you something (miles, points, cashback, whatever) everywhere, and then you use it for AAdvatnage Dining, you will earn the AA miles from AAdvantage Dining, and totally separately earn the whatever something (miles, points, cashback, whatever) that the card always gives you.

Thus if you use a hotel-affliiated card (except Priority Club*), you'll earn points for that hotel, plus the AA miles. If you use an airline-affiliated card, you'll earn the normal card miles for that airline (which can be AA or certain others*), plus the AA miles from AAdvantage Dining.

*The only restriction is that some airline credit cards, as well as Priority Club hotel credit cards (Priority Club is the frequent stay program for InterContinental, Crowne Plaza, Holiday Inn, Express, Candlewood, etc), you get automatically enrolled in the Rewards Network dining program for that airline or credit card, and you can't enroll one card in two different Rewards Network dining programs at the same time.

But in this example, since Hilton HHonors doesn't participate in a Rewards Network dining program, there's no problem using its card to earn HHonors points while earning AA miles. Similarly, Best Western, Choice, Starwood, and Marriott are among other hotels at which this would work, and Southwest and Frontier are among the airlines that it would work at (because those are airlines that don't participate in Rewards Network operated dining programs). And cashback (or card-specific rewards points) cards from Citi, Chase, Amex Blue, Amex MR, Discover, etc, would also work.

So which card to use with AAdvantage Dining depends on what else you most want to earn. If your whole goal is to maximize AA miles, then you might want to use an AA-earning card (an AA credit card, a Starwood AMEX, or a Diners Club). But all those have annual fees (except an AA Citi Bronze, but that only earns 1 mile/2$), while the hotel cards don't, so if there's a hotel chain you already stay at and you just want to top off those points (or keep them from expiring, in the case of HHonors which expires after a year of no activity), then a hotel points card might be better for you, especially if your main goal with AA is simply to keep existing AA miles alive.