There are a lot of hyperbolic trigonometric identities derived in Poincare disk model that resemble similar identities from the Euclidean geometry.
For example, the analogue of the Pythagoras theorem is $\cosh(c) = \cosh(a)\cosh(b)$, the analogue of the sine law is $\frac{\sin(A)}{\sinh(a)}$ = $\frac{\sin(B)}{\sinh(b)}=\frac{\sin(C)}{\sinh(c)}$ and so on.
After reading the proofs of those identities it made me wonder whether they are specific for the Poincare disk model, or they can be arrived at in other models, e.g. Beltrami-Klein model.
I suppose in other models those identities look differently, but I'm not sure. If so, how does Pythagoras theorem looks in Beltrami-Klein model?
The identities of hyperbolic trigonometry are true intrinsically. They are equations relating distances and angles of triangles (and other shapes), and those distances and angles themselves are intrinsic to hyperbolic geometry, so the identities that they satisfy are true independent of which model of the hyperbolic plane that you wish to work in.
Perhaps you are misled by the fact that some models of the hyperbolic plane are conformal, others are not. A conformal model means a model embedded within the Euclidean plane, such that hyperbolic and Euclidean angles are equal. The Poincare disc model is conformal, as is the upper half plane model. The Beltrami-Klein model is not conformal. But that doesn't matter because those models are all isometric to each other, meaning that between any two of those models there is a bijection that preserves the distances between points, and so everything derived from those distances (e.g. angles) is also preserved, hence all identities about distances and angles that are true in one model are also true in the other.
Perhaps what would be most convincing is to see how angles in the hyperbolic plane can be defined without using any particular model. They can be defined intrinsically, using only the concepts of distance, line segments, and rays, just as they can in the Euclidean plane. In fact, the definitions are more or less the same in either case. In brief, for each point $P$ the orientation preserving isometries of the plane that fix $P$ form the circle group $\left\{ \begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix} \right\}$; the angle between any two rays based at $P$ is the value of $\theta$ which takes one ray to the other under the action of the circle group.