Grassmannian as a quotient of orthogonal or general linear group

Answer to first question: there is a very often abuse of notation here. The block diagonal subgroup $\begin{pmatrix}A_r\\&A_{n-r}\end{pmatrix}$ is isomorphic to $O(r)\times O(n-r)$, and we are quotienting $O(n)$ by this subgroup. There could be other isomorphic copies of $O(r)\times O(n-r)$ which could give different quotients, but they are of no interest to us.

Let's also settle what quotienting does here. The quotient manifold theorem says

Suppose $G$ is a Lie group acting smoothly, freely and properly on a smooth manifold $M$. Then the orbit space $M/G$ is a topological manifold of dimension $\dim M-\dim G$ and has a unique smooth structure with the property that $M\to M/G$ is a smooth submersion.

In particular, the quotient of a Lie group by a Lie subgroup is a smooth manifold. You don't get a "metric structure" just from this information, although you could define a (noncanonical) Riemannian metric on patches arbitrarily and glue using a partition of unity.

Answer to second question: Note it is "$H$ is the stabilizer of any ...", not "$H$ is the stabilizer of all ...", so $H$ just stabilizes one $r$-subspace of your choice.