Your two final questions are not the same, in spite of the "That is,..." that starts the second one, but, in the interpretations that make the most sense to me, the answer to the first is 'no' and the answer to the second is 'yes'.
For the second question, you know that a smooth manifold $M$ has a compatible real-analytic structure and, by the theorem of Grauert and Morrey, it can be real-analytically embedded into an Euclidean space of sufficiently high dimension. Then the induced metric from such an embedding is real-analytic, so every smooth manifold does have a metric that is real-analytic with respect to some compatible real-analytic structure.
Moreover, when a smooth $M$ is compact (and maybe even in general), every smooth Riemannian metric can be approximated arbitrarily closely (in practically any sense) by a metric that is real-analytic with respect to some compatible real-analytic structure on $M$. You just use the given metric as the initial metric for the Ricci flow (which exists for short time $0\le t < T$), and the resulting metrics $g(t)$ for any positive time will be real-analytic with respect to some real-analytic structure (that is independent of $t$). Thus, each smooth metric $g$ on a compact $M$ determines a canonical real-analytic structure, and $g$ can be approximated arbitrarily closely by metrics $g(t)$ (for $t$ small and positive) that are real-analytic in that real-analytic structure.
Now, my interpretation of the OP's first question is this: Given a smooth Riemannian manifold~$(M^n,g)$, is there an analytic structure on $M^n$ in which $g$ is real-analytic? If one doesn't always exist, does one exist for a 'generic' smooth metric $g$ on $M$?
Well, it's certainly not true always. For example, if $n=2$ and $g$ is a metric whose Gauss curvature vanishes to infinite order at a point of $M$ but is not identically zero, then $g$ clearly cannot be real-analytic for any real-analytic structure subordinate to the given smooth structure. It's easy to construct such examples. For example, take $g = e^{2h(x,y)}(dx^2+dy^2)$ where $h$ is a smooth function on $\mathbb{R}^2$ such that $\Delta h$ vanishes to infinite order at $(x,y)=(0,0)$ but isn't identically zero.
Moreover, it's also not true 'generically', in the following sense: Again, in dimension $2$, we know that any smooth metric is conformally flat, i.e., each point lies in the domain of a smooth coordinate chart $(x,y)$ such that $g = e^{2h(x,y)}(dx^2+dy^2)$. Moreover, if $g$ is analytic in some local coordinate system $(u,v)$, then any conformal coordinate chart $(x,y)$ in that domain must be analytic as functions of $(u,v)$. Thus, $g$ is analytic in some coordinate system if and only if it is analytic in conformal coordinates. However, when you write the 'generic' $g$ in conformal coordinates $(x,y)$, the function $h(x,y)$ will not, in general, be an analytic function of $(x,y)$. Thus, the generic smooth metric in dimension $2$ is not real-analytic in any coordinate system.
Similar remarks hold in all higher dimensions. A 1981 result of DeTurck and Kazdan shows that a metric $g$ has its maximum regularity in $g$-harmonic coordinates, i.e., $g$ is real-analytic if and only if it has analytic coefficients when expressed in a coordinate system $x=(x^i)$ where $\Delta_g x^i = 0$ and where $\Delta_g$ is the Laplace operator associated to $g$. In particular, if $g$ is analytic in any real-analytic atlas, its system of $g$-harmonic local coordinates must have real-analytic transitions functions, and these belong to the only possible real-analytic atlas in which $g$ could be real-analytic.
This 'test', though, may not be entirely satisfactory because it requires one to find (local) solutions of a PDE to make a (local) coordinate system and then recognize whether the coefficients of $g$ in that coordinate system are real-analytic functions of the coordinates. There's no way to avoid needing a way to recognize when some function is an analytic function of some other functions, but one can usually avoid the step of solving a PDE.
For example, again, in dimension $2$, given a metric $g$, one can compute its Gauss curvature $K$ and the squared norm of its gradient $L = |\nabla K|^2_g$. Suppose that these are independent functions (i.e., local coordinates) on some open set $U\subset M$. Now compute $\Delta_g K$. If $g$ is real-analytic in some coordinate system with domain $U$, then $\Delta_gK$ will be an analytic function of $K$ and $L$. One can show that, conversely, if $\Delta_gK$ is an analytic function of $K$ and $L$, then $g$ has real-analytic coefficients in the $(K,L)$ coordinate system.
