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Gabe Conant
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First note that $M$ is countable while $M^*$ has size continuum. So the two structures are never isomorphic. But you may instead ask when/if they are elementarily equivalent; and the answer is “sometimes”.

Example 1: Let $\mathcal{C}$ be the class of finite graphs. Then $M$ and $M^*$ are both models of the theory of the random graph.

Example 2: Let $\mathcal{C}$ be the class of finite linear orders. Then $M$ is isomorphic to $(\mathbb{Q},<)$, while $M^*$ is a ($\aleph_1$-saturated) discrete linear order with endpoints.

Gabe Conant
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