I believe there is no good model of $\mathbb{E}^2$ in $\mathbb{H}^2.$ However, there is an excellent model in $\mathbb{H}^3:$ any horosphere will work.

I believe there is no good model of $\mathbb{E}^2$ in $\mathbb{H}^2.$ However, there is an excellent model in $\mathbb{H}^3:$ any horosphere will work.

Also This is not particularly interesting, but if you use the hyperboloid model of $\mathbb{H}^2,$ you can project it (from, e.g., the point $(2, 0, 0)$ onto the $(x, y)$ plane. This will give an algebraic model of $\mathbb{E}^2$ in $\mathbb{H}^2.$