In Classical Mechanics, a system has configurations described by points of a configuration manifold $Q$. In that setting we define the phase space of the theory to be the cotangent bundle $M = T^\ast Q$ which has the interpretation of being the space of pairs of configurations and associated momenta.
In local coordinates, if $(q,U)$ is a chart in $Q$ which gives rise to generalized coordinates $q^i$ we have on $M$ the chart $(q,p,T^\ast U)$ where we have the lifted generalized coordinates $q^i$ together with the conjugate momenta coordinates $p_i$.
Now let's take classical field theory, which should be the straightforward generalization to infinitely many degrees of freedom. For that let us consider a classical Klein-Gordon field $\phi$ on some globally hyperbolic spacetime $(M,g)$ with volume form $\epsilon$.
It is described by the action
$$\mathcal{S}[\phi]=\int_M \mathcal{L}\epsilon,\quad \mathcal{L}=\dfrac{1}{2}(g^{ab}\nabla_a \phi\nabla_b\phi+m^2\phi+\xi R\phi^2).$$
Here $\phi$ plays the whole of the coordinates. Furthermore, if we fix some coordinate chart $x^\mu$ where $x^0$ has the meaning of time, we can define the canonicaly conjugate momentum in analogy to the finite degrees of freedom case
$$\pi=\dfrac{\partial \mathcal{L}}{\partial(\nabla_0\phi)}.$$
In that setting, it seems that $\phi,\pi : M\to \mathbb{R}$ and that the phase space defined in analogy to classical mechanics would be the space of all pairs of maps $(\phi,\pi)$ like these, and which are related as above.
On the other hand, it seems that things are done differently. What I've seem done in a few sources is that one picks one Cauchy surface $\Sigma\subset M$ on which initial data can be given for the KG equation. The data is taken to be pairs $(f,g)\in C^\infty_0(M)\times C^\infty_0(M)$ so that
$$\phi|_{\Sigma}=f,\quad \nabla_0\phi |_{\Sigma}=g.$$
Then one takes the phase space $\Gamma$ to be the space of all such pairs of initial conditions.
Why is that? This doesn't seem straightforward from classical mechanics. What is the motivation to take this as the phase space, and how does this relates to the classical mechanics of finitely many degrees of freedom phase space?
