> Ok, so this feels like the crux of the matter. So how does the fact that V* is not isomorphic to V make the tensor product construction a more general concept than the linear function construction?
This doesn't really have anything to do with tensor products per se; it can already be seen with tensor products involving only a single factor, which are just vector spaces. Thinking of tensors only as functions means that, for example, one can never think of the original vector space V itself, only of its image in the double dual V^{**} (a fancy way of saying that you can evaluate a vector v \in V on an element v^* of the dual vector space V^* by evaluating v^* at v: in confusing but suggestive notation, v(v^*) = v^*(v)).
It's certainly true that you can do this, and, given the axiom of choice, you don't lose any information; you know everything about a vector v \in V by knowing its value on elements of V^* (which is to say, by knowing the values of elements of V^* on V). However, if V is infinite dimensional, then you are forcing yourself to carry around extra, possibly unwanted information: if you are taking bare algebraic duals, not topological duals, then V^{**} is inconceivably larger than V, which is to say that there are way more linear functionals on V^* than just those coming from evaluation at a fixed element of V.
You can fix some of this inconceivable largeness by knowing a little more structure carried by V, and by forcing your dual to reflect that structure—usually you know the topology, and ask that the dual consist of continuous functionals; and once there's topology, you start asking things of the tensor product, too. (For example, you probably don't want to take the vector-space tensor product of Hilbert spaces, but rather its completion in some suitable sense.) But, even with a more refined notion of duality, it's still only the nice spaces V that are identified with their double duals via the canonical map V \to V^{**}; the terminology is 'reflexive'.
(I think I caught all the asterisks that the Markdown parser ate the first time through.)