However, depending on the choice of the order for Iwasawa decomposition and on the choice of a left or right invariant metric on $SL(2,\mathbb{R})$, we get a twisted and a non-twisted (product) metric on $\mathbb{H}^2\times \mathbb{R}$.
The twisted one coincides with the metric on the hyperbolic plane, for certains levels of the fibers (slices).
The product one is a product, but the metric on the slices has nothing to do with the hyperbolic one...