Why are all circles similar? (Why is $\pi$ a constant?)

Up to translations, you may assume that all the circles are centered at the same point $p$. Now the existence of a dilation sending each circle into another is obvious, because the property "being equidistant from $p$" is invariant under dilations with center $p$.

They're not!!

Circles have circumferences which depend on the radius, except if you happen to be in Euclidean space (a space with zero curvature). For instance, in hyperbolic space: $$\frac{C}{r}=2\pi \frac{\sinh r}{r}\neq 2\pi$$

So really, the question is how we know (or the ancients knew) that in Euclidean space the ratio is constant. This can be seen by observing triangles, and noting that as long as the angles are the same on a plane, the side lengths scale linearly. Subdividing a circle into many small triangles, shows that this logic then must apply to circles as well - if you increase the radius, the circumference must scale by a constant factor, which happens to be $2\pi$.