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The concept of an isomorphism is very interesting: a rigorous, formal way of expressing similarity between two objects. When and how did this idea of similarity become formalised as a bijective mapping between two spaces?

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    $\begingroup$ There is no such thing as "theory of isomorphisms". The notion of isomorphism evolved gradually, beginning from prehistoric time, when people started counting objects, and taking its final present form with Bourbaki who defined structures and isomorphisms of structures. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 6, 2019 at 16:18

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I found this in Earliest Known Uses of Some of the Words of Mathematics

ISOMORPHIC, ISOMORPHISM. The term isomorphism was used early in crystallography. Some books on geology before 1864 referred to “geometrical isomorphism” or “mathematical isomorphism” between crystals.

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