plane geometry – What does it mean to say that a Riemannian metric is dxdy/y²?

This is a question about notation, so it may seem trivial, but I need a confirmation that my understanding is correct.

I came across a course on automorphic forms given by Bảo Châu Ngô at the College de France. At some point during the presentation he considers the plane with a riemannian metric defined by : $$frac{dxdy}{y^2}$$.

In order to understand what that means I need to relate it to something I understand and know how to calculate : euclidean metric. Is euclidean metric defined by : $$sqrt{dx^2+dy^2}$$ ?