Even though Culture isn't directly related to a culture victory now, it's still incredibly useful for downright everything, and the ability to amass loads of it puts you a step above the rest. For a culture victory especially, adopting Aesthetics and even Piety will help you greatly.
You can't Jaguar-spam very well now, this is true. The Aztecs have never been about unit-spamming, they have been about quickly and efficiently wiping out as many enemies as possible with as few units as possible. You want to be lean and effective with your units, thinking a step ahead to make sure you lose none and kill as much as you can. They've always played differently from others, in that they are most effective at cultural achievements by way of war policies and techs. You want to set your military up to be as effective as possible, and just slaughter everything you can all the time. If you're not attacking something every turn, you're doing it wrong. You need to constantly be killing something, at first kicking it off with wiping out barbs, and then, once they run out, declaring war immediately.
They play differently now, because growing massive cities is key to a culture victory as you absolutely need to work those guilds. The Floating Gardens is no longer simply a nice thing - it is the key to turning all that culture you generate into something useful. Getting massive amounts of food, combined with the bonuses Aesthetics can give, means you can easily pump out those Great Works at lightning speed.
One would quickly compare them to Poland, who gets a raw 7 free social policies over the course of the game, and I suppose it would be useful to crunch numbers and see if the Aztecs, on average, get more social policies at a typical rate than Poland does, but it doesn't work that way. Your culture from kills is a continuous stream and not a periodic event, which does entail different things obviously, namely that you'll be getting more early social policies than everyone else. It's very flexible now, and that probably defines the Aztecs best now. They aren't directly pushing towards a victory, but now that early-game culture requires more effort and isn't a matter of just building the buildings, since they only give a tiny bit of culture at base, the Aztecs have a definite edge, being able to force their way through multiple social policy trees early on, so long as you are just constantly at war. They are not a safe or easy civilization, they require wise planning, and playing a hybrid of defense and offence. You're investing in military, but not for the sake of taking other cities. To use them right, you need to make sacrifices that often won't pay out in the way the game intended. As a simple example, you are probably going to take Autocracy as the Aztecs. The ideology is set up to where you get mondo happiness to offset the penalties from taking other cities...but you're not taking other cities, not en masse. So that happiness is going more into Golden Ages than anything. In fact, in general I'd say they benefit a lot from how Autocracy has bonuses towards a culture victory, and I'd go so far as to say that's what they're meant for, as the bonuses therein fit them to a T. You can definitely pick any ideology you want, because you'll be generating the most raw culture out of everyone in the game, and that DOES go towards your defensive culture bucket. You don't have to worry about what others pick, because their tourism is not going to break you effectively, especially if you're making enemies of them. And you will be making enemies, because you have nothing to gain from being at peace. Your UU is completely irrelevant if you're not at war, but it never wears off, and grows stronger as the game progresses.