Unique metric for the Hyperbolic Half Plane Model?

First, note that the transformation rule defines an action of $GL_+(2, \Bbb R)$ (the group of $2 \times 2$ real matrices with positive determinant) on the upper half-plane, namely via $$\pmatrix{a & b\\c & d} \cdot z := \frac{a z + b}{c z + d}.$$ For any such transformation, if we scale each of $a, b, c, d$ by $\lambda \in \Bbb R - \{0\}$, the transformation is unchanged, and the quantity $a d - b c$ scales by a factor of $\lambda^2$; so, we don't lose anything by restricting our attention to the transformations for which $a d - b c = 1$, that is, by considering only the restriction of our action to $SL(2, \Bbb R)$.*

So, we're now looking for a metric invariant under the given $SL(2, \Bbb R)$-action on $\Bbb H$. In particular, it must be invariant under the isotropy action at any point. Direct computation shows that the isotropy subgroup at the convenient point $i \in \Bbb H$ is $$\left\{\pmatrix{a & -b \\ b & a} : a^2 + b^2 = 1 \right\} \cong SO(2);$$ the isotropy action thus preserves a unique inner product (up to scale) on the tangent space $T_i \Bbb H$, and some computation shows that $$dx\vert_i^2 + dy\vert_i^2 = dz\vert_i \,d\bar{z}\vert_i$$ is such an inner product (here, $x, y$ are the usual real coordinates on $\Bbb C$, namely the ones characterized by $z = x + i y$). Pulling this metric back by arbitrary elements $g \in SL(2, \Bbb R)$ gives that the metric $$\color{#bf0000}{\frac{dx^2 + dy^2}{y^2} = \frac{dz \,d\bar{z}}{(\Im z)^2}},$$ the usual hyperbolic metric, is invariant under $SL(2, \Bbb R)$.

Remark *Given such a transformation with coefficients $a, b, c, d$, the transformation with $-a, -b, -c, -d$ defines the same transformation, so the $SL(2, \Bbb R)$-action in fact descends to an action of $PSL(2, \Bbb R) = SL(2, \Bbb R) / \{\pm I\}$, and this latter action is faithful. It's often more convenient, though, to work with $SL(2, \Bbb R)$, whose elements are bona fide matrices, at the cost of a little redundancy.