A separable Banach space $E$ is $\omega$-categorical if its orbit space (space of complete types) $E^n / \mathrm{Isom}(E)$ is compact for all $n$. A related property is being approximately ultrahomogeneous (AuH), which holds when these orbits coincide with isometry types. We investigate certain $\omega$-categorical spaces. We provide a precise characterization of the orbit spaces for $C(K, X)$ (where $K=2^{\mathbb{N}}$) and $L_p([0,1], X)$, assuming $X$ is $\omega$-categorical. We show that $(C(K, X))^n / \mathrm{Isom}$ is isometric to the space of compact subsets of $X^n / \mathrm{Isom}(X)$, and $(L_p([0,1], X))^n / \mathrm{Isom}$ is homeomorphic to a space of probability measures on $X^n / \mathrm{Isom}(X)$. We also compute the Roelcke compactification of the corresponding isometry groups. These results provide a geometric explanation for the failure of AuH in $C(K)$ and $L_p$ for $p \in 2\mathbb{N}+4$. This is joint work with V. Ferenczi and V. Olmos-Prieto, and forms part of the latter's PhD thesis.