I think you are right to say that both homogeneous spaces are diffeomorphic as smooth manifolds. Therefore if one of them is a symmetric space, then one can transfer this structure to the other one.
However it is common usage and an abuse of language to say that $G/K$ is a symmetric space to mean that $(G.K)$ is a symmetric pair, that is, $G$ is a connected Lie group and $K$ is an open subgroup in the fixed point set of an involutive automorphism of $G$. This is essentially equivalent to your Lie algebraic formulation. In the strict latter sense, $SO(2n+1)/U(n)$ is not a symmetric space.