Picture below is from the 167th page of do Carmo's Riemannian Geometry. I don't know the mean of "the isometries that identify the sides of $P$". My English is poor. I know the process of identifying the sides of polygon to get a surface. For example, gluing the sides of rectangle by a certain way, we can get the torus. But I don't know the relation between the gluing and isometry.
First, the isometry will map the internal point of $P$, but the gluing will not affect the internal point of $P$.
Second, the isometry that identify the sides of $P$ is that $$ I:H^2\rightarrow H^2 ~~~~~~\text{is isometric, and } ~~~~~~I(\partial P)=\partial P ~~~~\text{?} $$
