That's a good question.

The tl;dr is that if information hides on the other side of an event horizon and doesn't come back, we can pretend unitarity (and all the rest of the physics we've discovered) continues where we can't see it.

A non-evaporating black hole forever holds within it the information about what fell into it.

A forever-expanding universe causes information to exit observability forever.

Partitioning away -- hiding forever -- information is not the same as losing track of it when it comes out of hiding.

There are some differences between these two types of horizon because they are generated by different metrics : one for an expanding spacetime and one for a collapsing one.

We can see the differences by adapting these theoretical (as opposed to astronomically observed) objects.

If an expanding universe's expansion slows and reverses, then eventually all the galaxies that exited from one observer's view return into its view (having evolved with stars forming, aging, dying, galaxies merging, and so forth). If we are talking timescales of a few billion years, then if an us-like observer has detailed information about a galaxy now leaving its view, it can in principle predict what it will look like in billions of years when the galaxy returns back into view. The gentle assumption here is that stellar physics does not stop when the most distant galaxies go out of view.

If the timescale is pushed out to trillions and zillions of years, these us-like observers could still maintain the idea that the galaxies which exited from visibility continue to evolve like the closer galaxies which continue to be seen. A star which ends up on the other side of the cosmological horizon continues being that same star, evolving as normal.

A black hole is different, precisely because we should expect unknown extremely high energy physics to occur as e.g. protons fall in. What happens as you crush some quarks and gluons together at energies enormously higher than that we get from the LHC, or even from supernovae? We don't know. In fact, when we try to answer that, we lose track because our calculations tend to become singular : https://en.wikipedia.org/wiki/Singularity_(mathematics) We don't know what should pop out of a black hole late in evaporation, but we do know when a star crashes through a black hole event horizon, it will stop behaving like a star very quickly.

Indeed, even just on "our" side of the two horizons we can see differences near them. The furthest galaxies, at the edge of what we can see of the cosmos, are filled with normally behaved (young) stars. The shapes of those galaxies are not distorted by proximity to any horizon. We expect that to continue as we see galaxies deeper and deeper into our sky. By comparison we can see gas clouds falling into the black hole in the centre of our galaxy, stars orbiting it, and distortions to these caused by these close approaches to the central black hole. We have even found evidence of stars ripped apart by more distant extragalactic black holes. Crossing a black hole horizon does violence to the bit of the star that has not yet crossed; crossing the cosmological horizon would not change the star's basic behaviour.

If the universe were to collapse in the future, we would expect to see disappeared stars returning into view. Those stars stayed in locally gently curved spacetime, just like our local star did. If a black hole were to shrink in the future, we would be surprised if it spat out intact stars, or space probes, or whatever fell in emerging unscathed. Those objects did not stay in locally gently curved spacetime, and indeed would have encountered the locally enormously curved spacetime inside the black hole. That strong curvature spaghettifies things, at the very least.

These are just the consequences of our best theories of gravitation and matter applied to situations we have no reason to expect to be able to observe. As far as we know our universe is not accelerating towards a recollapse, it is accelerating towards faster expansion. And as far as we know no astrophysical black hole in our universe is presently shrinking. It's fairly safe to bet that if there is ever to be a reversal of the expanding cosmological metric or the collapsing black hole metric it's not going to be soon, so humanity and its descendants have lots of time to think about evaporating black holes (including those that evaporate in a contracting "anti-de Sitter" universe with a big crunch, which is the setting (sometimes including extra spatial dimensions than the three we're used to) for many approaches like the one in the fine article in Quanta Magazine linked at the top).

Now, a more direct answer to your question: in an almost-completely-flat-space universe if we have all the data (position, momentum, particle species, etc) at every point in a time-indexed spatial slice of our universe, we can calculate the entire data in neighbouring slices, and the data in those slices' neighbours, and so forth, into the infinite future and the infinite past. An expanding universe doesn't break this, it just means that we can't choose any arbitrary slice and march forwards and backwards from there, we have to take initial data from the hottest densest earliest part of the universe. From complete initial data and appropriate dynamical laws we can (in principle) describe anything in the future, even if the parts we describe are so separated from one another (in that future) that they can't exchange light with one another. The formation of non-evaporating black holes doesn't change the picture much: we know that things fall into a black hole and stay there in some unknown state, unable to exchange light with things outside the black hole.

However, once we introduce black hole evaporation we have the problem that we don't know how the stuff inside the horizon evolved inside the horizon, so we have no idea what should pop out through the last stages of evaporation.

In our standard cosmology, we can expect black holes to have evolved from stuff that was close to us in the hot big bang era but which is now almost certainly forever outside our cosmological horizon. A general solution to the black hole information paradox should not create craziness in those so-distant-we-will-never-see-them black holes, much less in the earliest visible quasars. That tends to get forgotten until someone asks what the interviewer asked:

Most of the justification for the quantum extremal surface formula comes from studying black holes in “Anti-de Sitter” (AdS) space — saddle-shaped space with an outer boundary. Whereas our universe has approximately flat space, and no boundary. Why should we think that these calculations apply to our universe?

That's an excellent question, and it was not answered by the interviewee. (I'd love to be persuaded that it has ever been reasonably answered by anyone).