It's confidence = trust, belief, faith (uncountable abstract mass noun) in OP's OALD citation, not the count noun secret, private communication sense as subsequently quoted.
The every confidence usage is something of an oddity. Note this from an 1869 grammar book...
We should not say We have every confidence in him, because confidence is an abstract noun, and does not mean one of a class of objects. The expression should be I have all confidence in him.
But as every learner should be aware, native speakers don't always respect the "logical" pronouncements of grammarians, and that one has always been largely ignored. More recent "descriptive" grammarians now define the usage as...
Typical uses of every (Pocket Fowler's Modern English Usage, 2nd Ed., 2008)
2 - with some abstract uncountable nouns referring to a feeling or attitude: [italics mine]
We have every sympathy for their case
I have every confidence in you
Almost exactly the same description of this "valid" usage appears in Harrap's essential English Dictionary, 1996, which specifically targets the needs of Indians learning English.
It's purely my own opinion, but personally I feel this is something of a "post hoc rationalisation" for usages which might really be better described as "idiomatic, fixed expressions". It's also worth noting that the confidence version has held up far more than the sympathy version, which I feel reflects the fact that all such usages are tending to fall into disuse (or at least, they're perceived by native speakers as being somewhat "quirky, old-fashioned, formal").
My advice to learners would be to accept...
have every ["feeling/attitude" abstract noun] in/for/to [the target of that feeling]
...as "valid", but to avoid extending the usage. There's no obvious way of identifying which abstract nouns everyone is happy with (confidence, success), as opposed to more "marginal" versions (sympathy, faith), and completely unacceptable forms (such as ∗I have every love for her).
In practice I imagine that if they think about it at all, most native speakers perceive every in such contexts more as an "intensifier" (I have complete/absolute/total confidence in him), rather than a "quantifier" (I have every/all/many reason/s to trust him).