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I encounter the terms "world" and "universe" in various types of philosophy. I haven't paid close attention, but it seems "world" is used more phenomenologically and hence in Continental genres, where "universe" is used in physics as....I don't know, maybe the unity of a Newtonian spacetime of "all countable things." I really don't know.

So, my question is, are these formally defined in analytical philosophy? Which is the branch of philosophy that does formal definitions. I mean, is there some formal definition or some general distinction? And is there a difference in physics and Anglophone philosophy (not sure what else to call it now) between "many worlds" and "multiple universes," where the latter certainly seems self-contradictory.

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    Sidestepping any call for univocal defintions, the definitions and meanings of both terms are applied flexibly. At times and in certain contextual frames they are used synonymously. That usage can readily be picked up in the authors stylistic use of language. At other times world is used as you suggest from the perspective of the Observer, as in world view. Universe, in traditional metaphysics typically involves an attempt to capture and describe the totality of all possibility within micro/macro evolution and recreation. And yes, this not what you are looking for, but enjoyable anyway! Cheers – user37981 Oct 24 '20 at 3:14
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    I do not think there is a formal definition or coherent distinction, the terms are used opportunistically, e.g. "possible worlds" vs "universe of discourse" simply come from historical precedents. In physics "a world" is usually something smaller than "the universe", the entire spacetime or its spatial slice, "other worlds" can be other planets or planetary systems. Multiverse threw a wrench into this size distinction by relativizing "the universe" as "all there is". The "actual world" is now analogous to the universe, and the class of all possible worlds to the multiverse. – Conifold Oct 24 '20 at 7:46
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    See World – Mauro ALLEGRANZA Oct 24 '20 at 16:30
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    World understood as " cosmos" refers to a totality organised hierarchically . Universe refers to a totality unified by laws so to say egaliarian, applying everywhere in a uniform way ( galilean universe wihout any distinction as to the laws governing heavenly bodies and sublunar bodies). – Floridus Floridi Oct 26 '20 at 10:07
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    Reference : Koyre , persee.fr/doc/rhs_0048-7996_1958_num_11_4_3678 – Floridus Floridi Oct 26 '20 at 10:14

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