I suppose you're referring to statements in hyperbolic dynamics of the type "the angle between $E^s(x)$ and $E^u(x)$ is bounded away from zero".

I haven't seen a formal definition, but I think the typical implicit assumption is that a Riemannian metric is chosen, and under this metric the minimum angle between tangent vectors $v \in E^s(x)$ and $w \in E^u(x)$ defines the angle between $E^s(x)$ and $E^u(x)$.

This depends on the choice of metric on $M$, but if for a metric $g$ this angle is bounded away from zero, uniformly for all $x \in M$, then the same holds true for any other metric $g'$, since any two Riemannian metrics are equivalent on a compact manifold in the sense that there exists a constant $C > 1$ such that
$$
C^{-1} g'_x(v,w) \le g_x(v,w) \le C g'_x(v,w)
$$
for all $x \in M$ and $v,w \in T_x M$.