Yes! It can happen and is quite important.
The Teichmüller space of a Riemann surface provides a counterexample to your discreteness assertion. See, eg. Tromba's book, Teichmüller Theory in Riemannian Geometry, p.10. This space encodes the set of inequivalent complex structures on a fixed Riemannian manifold and positive dimension as soon as the genus is greater than 1.

Even simpler, for genus 1, the two-torus, we know that the set of moduli of complex structures is identified with the modular surface (upper half-plane)/$Sl(2,{\mathbb Z})$ which has complex dimension 1. To be explicit, think of the standard torus as ${\mathbb R}^2/{\mathbb Z}^2$ endowed with the symplectic structure $\omega_0 = dx \wedge dy$ where $x, y$ are standard coordinates for ${\mathbb R}^2$. Consider the usual description of the moduli space of elliptic curves, eg. from Ahlfor's text on complex analysis. The Teichmuller space is the upper half plane
and a point $\tau = \tau_0 + i \tau_1$ in this plane (so $\tau_1 > 0$)
gets sent to the elliptic curve ${\mathbb C}/{\mathbb Z}1 \oplus {\mathbb Z} \tau$. The complex structure on this curve is induced by projection from the standard complex structure on ${\mathbb C}$. Let $A$ be the real linear invertible shear which is the identity on the x axis and maps the unit y vector ${\partial}_x$ to $(\tau_0, \tau_1)$. Then $A$ defines a diffeo.from our standard torus onto this $\tau$-torus. Pulling back the $\tau$-complex structure
using $A^{-1}$ we get a new complex structure $J_{\tau}$ on our standard torus given by $J_{\tau}(\partial_x) = e_{\tau}$ and $J_{\tau}e_{\tau} = -\partial_x$
where $e_{\tau}= (-\tau_0/\tau_1)\partial_x + (1/\tau_1) \partial_y$. The $J_{\tau}$ form a continuum of inequivalent complex structures on the standard torus. (What about the metric condition? I compute the $\omega_0 ( \cdot, J_{\tau} \cdot)$ metric to be $(1/\tau_1) dx^2 + (\tau_0/\tau_1) dy^2$, a little wierd. I would've expected a quadratic form which is positive definite iff $\tau_1 > 0$, not iff both $\tau_0 , \tau_1 > 0$... . Maybe someone else can shed light on this (?))

To understand the higher dimensional version of this construction, look up the buzzword is Abelian varieties.

up to isomorphism. And in the case of $P^1$, $\mathcal{I}_int=\mathcal{I}$. In general, $\mathcal{I}_int$ is invariant by the whole symplectomorphism group of $(M,\omega)$, so it is infinite dimensional once it is nonempty (the stabilisers are finite dimensional -- riemannian isometry groups). $\endgroup$