Hadrian, did you ever work this out? I've been wondering lately if there's a Milnor-Moore over the sphere spectrum or some localization.

Yeah I agree. I know what "anima" means for sure. That's a straightforward terminology substitution. But there's a whole, like, renaming schema here, haha.

I think it would help if there were a little bit of context. There are a lot of people that know about this stuff, but they don't use the terminology "animated." You won't even tell us what conditions the categories need to satisfy to be "animated!" I'll try not to be cranky about new terminology, but it's not great when it gets hard to even communicate because people have completely abandoned the usual terms.

There is work of Angeltveit that computes the topological Hochschild coholomogy of K(n) (for certain versions of K(n), as Maxime points out) to indeed be Morava E-theory, just in case you want some kind of reference (even if vague). I don't recall which paper specifically. Perhaps this is discussed in Hopkins-Lurie though.

It's maybe worth mentioning that in the ∞-categorical context (and probably elsewhere, I just don't know of it) there is a notion of "universal coendomorphism object," and so a map of coalgebras Coend(X)→C is indeed a coaction. See, for instance, Aras Ergus' thesis here: aergus.net/academic/documents