It would appear that this may be a case of a paraphrase which became more popular and replaced the actual quote in our internet copy and paste, hallmark-quote based consciousness.
Researching online can find the quote used in several books, not just quotation sites. The first one I looked at was Hitler's Private Library: The Books That Shaped His Life by Timothy W. Ryback :
Benjamin proposed that a private library serves as a permanent and
credible witness to the character of its creator, leading him to the
following philosophical conceit: we collect books in the belief that
we are preserving them when in fact it is the books that preserve
their collector. "Not that they come alive in him," Benjamin posited,
"It is he who lives in them."
Note the quotations only on a certain portion of this text. Unfortunately no citation as to the source here.
Another work also contains your quote, The Perils of Print Culture: Book, Print and Publishing History in Theory and Practice by Jason McElligott
Walter Benjamin recognized that we collect books in the belief that we
are preserving them, when it is actually the books which preserve the
collector, his mindset, his milieu and the society that he inhabited.
This work actually included a footnote, citing the source, which is, as you suspected,
'Unpacking my Library: A talk about Book Collecting', in Walter
Benjamin, Illuminations (Frankfurt am Main, 1955; 1999 edition by
Pimlico of London).
A version of Illuminations can be borrowed from Archive.org, and the only portion of actual text I find therein is that which was directly quoted by the first source. From the final paragraph of the essay (pg. 69, emphasis mine):
For inside him there are spirits, or at least little genii, which have
seen to it that for a collector - and I mean a real collector, a
collector as he ought to be - ownership is the most intimate
relationship that one can have to objects. Not that they come alive
in him; it is he who lives in them. So I have erected one of his
dwellings, with books as the building stones, before you, and now he
is going to disappear inside, as is only fitting.