The Poincaré disk model of the hyperbolic plane is the open disk ${\rm int}(D^2)$ with a certain metric $d_H(x,y)$.
What happens if I take the open tube ${\rm int}(D^2)\times\Bbb R$ with the metric: $$d((x_0,x_1),(y_0,y_1))=\\\sqrt{d_H(x_0,y_0)^2+(x_1-y_1)^2}?$$ Does it have a name?
It is $\mathbb{H}^2 \times \mathbb{R}$ geometry.
In general, given two metric spaces $X$ and $Y$ with metrics denoted $d_X$ and $d_Y$, the product metric space is defined as a set to be the Cartesian product $X \times Y$ with the metric $d_{X \times Y}$ which, for each $p=(x_0,x_1)$ and $q=(y_0,y_1) \in X \times Y$ is defined by the formula $$d_{X \times Y}(p_0,p_1) = \sqrt{d_X(x_0,x_1)^2 + d_Y(y_0,y_1)^2} $$
There is a twist, however. What is more subtle is to take the Riemannian product metric, not just the metric space product metric. Thus, the above formula should really be used on an infinitesmal level, to define a Riemannian metric on each tangent space $$T_{(x,y)}(X \times Y) = T_x X \times T_y Y $$ But as it turns out, for complete Riemannian manifolds the two ways of taking the product are morally equivalent: the metric space metric $d_{X \times Y}$ is the same as what you get by minimizing path length using the Riemannian product metric.