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.