I am currently studying The Princeton companion to mathematics.
According the book, "A more precise definition is that the open unit disk is the set of all points $(x,y)$ such that $ x^2 + y^2 < 1$ and that the Riemannian metric on this disk is given by the expression $(dx^2+dy^2)/(1−x^2−y^2)$. This is how we define the square of the distance between $(x,y)$ and $(x + dx,y + dy)$." If the Riemann metric for the Poincare disk is just directly defined as $(dx^2+dy^2)/(1−x^2−y^2)$ without any proof, why can't we randomly use any expression as the Riemann metric? Is there any reason for choosing this specific notion of distance?
Also can there be multiple Riemannian metrics for a given manifold? How can there be multiple metrics for the same manifold?
Yes, given an underlying smooth manifold, there are generally many choices of metric. For instance, $dx^2 + dy^2$ (the standard Euclidean metric) is another perfectly acceptable choice. In this sense, there is no single "correct" metric to put on a particular smooth manifold, and it is just a definition.
However, if you give the "same" smooth manifold two different metrics, the results will probably not be the same as Riemannian manifolds. That is, while they have the same smooth structure (they are diffeomorphic), in terms of properties of the metric, they are different (they are not isometric in general).
Now, that said, we might arrive at the same Riemannian manifold via two different constructions. That is, we might start with $M_1$, give it a metric $g_1$, consider a distinct manifold $M_2$, give it some other metric $g_2$, and then show that $(M_1,g_1)$ is isometric to $(M_2,g_2)$.
This is useful, because you may be able to establish facts and properties about one model more easily than you could with the other. In the case of hyperbolic space, one such realization is the Poincare disc, but one can also construct the hyperbolic plane by, e.g., starting with a half-plane (instead of a disc) and a particular metric (and of course there are even more ways to construct this manifold that I won't list here, but look here if you are interested).