To identify an element of $SO(3)$, i.e. a rotation in $\mathbb{R}^3$, we must specify a direction in space, i.e. a straight line through the origin, and a rotation angle.
To specify an element of $SO(2)$ we just need the rotation angle.
Then one might argue that the quotient $SO(3)/SO(2)$ corresponds to the space of directions in $\mathbb{R}^3$, i.e. a projective space. But this is not the case: the quotient is actually $S^2$, the spherical surface.
Why must we not identify antipodal points of $S^2$? Do they not correspond to the same direction?
Your assumption that “To identify an element of $SO(3)$ […] we must specify a direction in space, i.e. a straight line through the origin, and a rotation angle” is not correct. To see why, consider, say, the straight line $\Bbb R(0,0,1)$ and the angle $\frac\pi2$. Does this describe an element of $SO(3)$? No; it applies to two elements of $SO(3)$, since there are two distinct rotations with angle $\frac\pi2$ around that line. So, it is indeed natural that $SO(3)/SO(2)$ can be seen as $S^2$.