It is always easier to think about the spherical case first. You can see that all round, two-dimensional spheres, regardless of radius, admit tessellations. So constant curvature $+1$ is not necessary. On the other hand, constant positive curvature should be a requirement. (In fact, it is fair to require that the metric be homogeneous and isotropic. Otherwise what does it mean for tiles of the tessellation to be identical?)
The case of constant negative curvature is the same. The choice of constant doesn't really matter, so you might as well use $-1$. For an elementary discussion, with many beautiful pictures, see "Noneuclidean tesselations and their groups", by Wilhelm Magnus. A more modern treatment, also with wonderful graphics is "Indra's Pearls" by Mumford, Series, and Wright.
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