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ubiquity (n.) [...] + que "any, also, and, ever," as a suffix giving universal meaning to the word it is attached to, from PIE root * kwe "and."
Did Etymonline err? I know that -que is an enclitic, and not a word.
Wiktionary attests -que to mean only 'and', as does AHI for the PIE root * kwe.
-que has generalising force only after pronouns and adverbs, e.g. ubique 'wherever', quisque 'whoever'.
I looked it up in Pons online Latin-German. There are so many translations of -que there, including some in which and simply doesn't work at all as a substitute, that any dictionary that presents and as the only translation must be considered incomplete.
Also is essentially a synonym of and when connecting two sentences. Since -que can be used in this way, it is not at all surprising that also made it on the list.
Fdb has explained where the translation -ever comes from. This explanation could also justify any-, since ubique can also be translated as anywhere. There may well be a better separate explanation for that, but I haven't found one.
In early PIE the full form of the clitic was -q̆et, it is a root meaning "a pair", other words with it include
q̆eta̯ pair
q̆etea̯ti counts
q̆etu̯ores four
This is a very ancient root, which meant quite the same (a pair) as early as Proto-Eurasiatic.
