According to Lewis, possible words are indiscernible from each other, that is, an individual in a possible world cannot say in which world it resides. It's somewhat similar to the indiscernibility of inertial frames of reference or to the indiscernibility of models of natural numbers created using ultraextension. My question is: does it follow merely from the fact that possible worlds are causally isolated or are there some other necessary conditions?
This is not quite right. Individuals in Lewis's worlds are world bound, in other words, each one only appears in a single possible world. In this sense, Lewis is the polar opposite of Kripke, whose individuals can be rigidly designated and appear in multiple worlds. In Lewis's modal realism Kripke's rigid designators, even proper names, will fail to refer except in a single world.
Perhaps what you mean is the problem of transworld identity for external observers. It is hard to pick the right referents in different worlds considering that we have only outward characteristics to go by, and there may be "twins" or "duplicates". Lewis resolves this problem in an unexpected way: there are no "right" referents, or rather all similar enough copies are right enough. So much so that for the purposes of reference there may be multiple counterparts of me even in the same world as me.
This is based on Lewis's interpretation of counterfactuals, sentences like "it is not possible for me to speak Finnish right now". How can we interpret this without transfering individuals into other possible worlds? His solution is the counterpart theory. Such counterfactual is true if "someone similar to me", my "counterpart", can speak Finnish right now. For Lewis this "similarity" is highly contextual, and the same someone may count as counterpart in one context, but not in another.
For more see Wallace's summary of modal realism:"To borrow one of Lewis’ examples, it is not possible for me to speak Finnish right now, since I don’t know the language. But it could have been that I learned Finnish when I was younger, or that I had just finished Intensive Finnish for Philosophers-in-Training, in which case it is possible for me to speak Finnish right now... Part of what contributes to the fact that our modal intuitions are so flexible from context to context in this way, Lewis thinks, is the fact that the counterpart relation is based on similarity relations, and such relations are highly flexible and malleable sorts of things.".
Indiscernability does not require more necessary conditions. In fact, it requires only one aspect of the causal isolation you are presuming.
Frames of reference are not causally isolated, but they are already indiscernable. To the degree that I still can't tell the world where I am in absolute motion from the world in which I am at absolute rest it must be the case that causal isolation is not a requirement. (I can frame this in terms of places in our world, or in terms of possible worlds, with equal meaning.)
What 'World identity' has in common with the other two phenomena you note is that the situations are absolutely relative. Identifying a privileged world is impossible because no such world exists, just as there is no 'right' aleph-3 cardinality model of the real numbers and no absolute resting frame of reference.
This proceeds from only a part of the causal isolation: that any causal interactions must be completely relative and not directed.
We can affect one another in relative frames, as long as there is a causal 'loop' or reversibility, that allows each of us to attribute the cause separately. We can both attribute it to a third party, or we can attribute it to one another. Since all cross-frame causes are reversible and not directed, we cannot follow the causes back to a privileged frame, and we lose the ability to determine which inertial frame we are in.
Extended models have the same sort of problem, where causal traction is removed by projecting your results onto another interpretation of the same model so that you cannot tell you were not already working in that model.
Of course having no causal links at all implies that no causal links are directed. But only the latter is a necessary condition to establish the relevant lack of identifying attributes.