The way I see it, the distinction between = and ≡ isn't really to do with equality having more than one meaning. An "identity" like sin^2 x + cos^2 x ≡ 1 is really just shorthand for "for every real number x, we have sin^2 x + cos^2 x = 1". The equals sign here has the same meaning as in a statement such as "there is a real number x such that sin^2 x + cos^2 x = 1"; the difference is in the surrounding language, and what meaning it assigns to the variable x.

So perhaps rather than just emphasizing the difference between = and ≡ more, it would be better to go further and emphasize the difference between universal and existential quantification more. Quantifiers can be confusing, but I think people also find having two different equals signs confusing; and it wouldn't be necessary to give a full account of predicate logic to high schoolers, I'm thinking more of just informally describing the difference between "for all" and "there exists" and reminding them that a bare variable has no meaning if you don't know what set it ranges over and how it's quantified.

This is just my speculation, I have no experience with mathematical education whatsoever.