I've never read Haeckel, but if he actually claimed this general principle was an invariant law, and if he claimed the gill slits were actual gills, then I guess he was wrong.
I have read (some of) Haeckel - he sadly mixed a true statement of the relation with what he called (IIRC)
The Fundamental law of Ontogeny which is a literal and strong statement of Ontogeny recapitulates Phylogeny and is false.
Haeckels admits (in response to his critics) in the forward to the second edition that his drawings are not photographically accurate. But maintains that they are diagramatically correct in a manner equal and similar to the manner in which any diagram is correct.
I think he is broadly correct in this claim; while his diagrams do omit some details and emphasise others they do not do so to an unacceptable degree and the features he draws attention to do exist.
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