It is a well known theorem: when a Lie group $G$ acts smoothly and transitively on a connected manifold $M$, for a fixed $x\in M$, there is a Lie subgroup $H \subset G$ defined by the group of elements $g\in G$ such that $gx=x$. It is possible to prove that changing $x\in M$, $H$ does not change. The theorem states that $G/H$ is diffeomorphic to $M$.

The fact that $H$ is (or is not) a normal subgroup is irrelevant here, since we are interested in the differentiable manifold structure of $G/H$. If we also want that this manifold receives a Lie group structure from $G$, we have to require that $H$ is a normal (Lie) subgroup of $G$.

Notice that a given manifold can be realized as different quotients $G/H$ if there are different groups acting smoothly and transitively on it.

In the considered case $SO(3)$ acts smoothly and transitively on $S^2$ and $H= SO(2)$.

So, it is not so simple to compute the quotient, because the theorem works along the other direction: One starts from a manifold equipped with a group of transformations and realizes the manifold as a quotient.

However the quotients of physically relevant Lie groups are computed. See for instance

R. Glimore, Lie Groups, Lie Algebras, and some of Their Applications

A.O. Barut R. Raczka, Theory of group representations and applications.