How is $dx^2+dy^2$ the Euclidean metric on $\mathbb R^2$

Question

A Riemannian metric is a smooth symmetric covariant $2$-tensor field.

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If I put in two vectors, say $(1,2)$ and $(2,1)$, I don't get $\|(1,2)-(2,1)\|=\sqrt{2}$:

$$(dx^2+dy^2)((1,2), (2,1))=2+2=4.$$