Wouldn't the word "passage" suffice? (That isn't enough of a reason for "passageway" not to exist, though.)
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From logic and memory I'll suppose that passage stems from the same root as, for example the English verb to pass, and French passer.
Which almost certainly implies that passage originally would have simply meant physical movement. The passage of time, a passage in a book, etc., would just be later extensions to meaning, based on metaphorical usage.
Thus passageway (doubtless originally two words) would have been coined to mean the actual path, corridor, or whatever through which passage took place. In practice, people often don't bother making a distinction between the movement itself and the route thereof, so passage often gets used for both / either.
Unlike EL&U, which formally tries to identify and weed out duplicate questions, the English language itself doesn't have a design committee to weed out words we could survive without. Anyway, I like having the choice of which to use.
It is a less ambiguous than passage which can be, for example, the 'passage of time' or a passage in a book. A passageway always means a physical space. If the context is ambiguous then 'passageway' may clarify it.
Obviously, synonyms will exist and two being similar to each other isn't really too surprising. "Way" seems to take it to an extreme, however:
The practice of using -way to more directly imply a path or location of travel is ridiculously common:
And the list goes on. Etymonline has some good comments on them. Many of the group are flagged as American English but the dates extend past the birth of the colonies.