What is the name of this logistic variant of TSP?
The problem is in $\mathrm{P}$ if the number of items is constant.

Let $K$ be the number of items (independent of $n$). For every ordering of items, use backtracking to try all allowed routes: you first go through some warehouse for the first item (trying all warehouses), then one for the second item and so on.

There are $O(K!)$ orderings of the items. Let $W_i$ be the number of warehouses for item $i$. The number of routes is $\prod_{i=1}^K W_i \leq \prod_{i=1}^K n = n^K$. Therefore, the running time of the above algorithm is $O(K! n^K)$, which is polynomial for fixed $K$.

If the number of items can be linear in $n$, the problem is at least as hard to approximate as $TSP$: you can take an instance of $TSP$, make an item for every vertex as you noted, and then add extra vertices very far away from all other vertices to inflate $n$ (and therefore allow for enough items that every vertex of the $TSP$ instance has a different item), without destroying the hardness of approxability of $TSP$. Note that if your points are in the Euclidian plane, then this doesn't really help you as there is a $PTAS$ for planar $TSP$.