Civ 4's relevant economies are Cottage, Specialist, and Wonder.
Civ BERT's Generator Economy is, at the moment, dead in the water. I've yet to try a Generator game but I suspect it won't run very well. There's just not that much synergy in it. Energy is useless in itself. You have to funnel it into something that allows you to win the game. It's not like Civ4 where you can just turn tile output directly into Science on your Slider.
Academy/Generator is closest to Civ4's Cottage Economy based on tiles and tile output - though the Generator part isn't necessarily since there's a bunch of other ways to make energy.
Specialist Economy in Civ4 relies on food discounts and Great People to equalize with Cottage Economy. Specialist Economy in BERT just relies on sheer slot output, no GPs. Food output is on the table in Mass Digester and Cloning Plants, so unlike Specialist Economy in Civ4, you don't even need food on the tiles or to make farms. You just need to build the right buildings in the right places (food buildings on land cities, Culture slots in Aquatic Cities).
Dome Economy is unlike anything in Civ4 since it puts Culture output in the spotlight. It's most like Culture strategies in Civ5. Dome economy focuses on Domes and Holomatrices, allowing you to claim and defend territory strongly, while getting a lot more Virtues constantly. It naturally pairs best with Purity's low tech Victory conditions.
ARC's (and Chungsu's) Spy Economies are unique in that you put everything into the spying. You don't even care much about your own Science output. It's all in the spies. Percentage science gains means that all tech costs about the same to you, regardless of their location, so you're cost-agnostic about science acquisition once you get something like 6 or 7 Agents. Of all the Economies, this one is the most likely to actually acquire Aegis, if you're into that unit for some reason. Annoyingly, the inner ring techs are kind of a "waste" of your Spies, but it turns out that many of those effects are kind of important to a Civ.
Polystralia's pure TR Economy depends on simply growing ASAP and sending out as many external TRs as possible while maintaining a strong internal system. A significantly amount of your science and energy output will come from external routes, so someone declaring war on you when they were your erstwhile ally and trade partner is nothing short of completely disastrous. At the same time, you can strengthen an ally and also forge better relations because of the number of TRs you forge. Provided that your external TRs are in-continent with a strong and trustworthy ally, it can be profitable to ally up and risk wars just for the extra tier of Agreement benefits.
INTEGR's DC Economy is all about attracting Agreements, keeping the peace and buying everything with Diplomatic Capital. You'll still need some way to make Science, but a few Institutes and Xenonurseries can put paid to that, even without Barre's benefits. You won't need much Energy since you'll be using DC for a lot of buys.
Interestingly, Daoming's Wonder ability isn't strong enough for her to base an economy around her sponsor trait, because the Wonders aren't strong enough to change the game in general, especially the early Wonders. Arguably, she benefits the most from a VF (Vertical Farm) approach since the best early Wonder she can snag from her ability is Benthic Augur, which pairs nicely with Euphrotic Strand from Vertical Farming - she has more cause than most to go down there.
The upshot of all this is that mechanically, BERT has some callbacks to Civ 4, but it's not really that similar.