I was taught the tensor product via its universal property: the only object that satisfies ... up to isomorphism. Later, I literally discovered one could actually write down the elements of (some) tensor products (I was asked to list the elements of the tensor product of two finite fields). It happened in France, because in my (Italian) university, no one ever showed anything like that to me.
Now I was wandering: one may introduce the definition of cartesian product through universal property, in a similar manner (I'm thinking about something like this).
However, wouldn't such definition be simply too difficult to understand as an undergraduate student (and depending on his level, even for a graduated one). Moreover, I would feel confident in saying that the real fulfillment of the need of knowledge for what a cartesian product is must come from the notion of "the set of all possible ordered pairs...", and only after it can be systematized in a culturally more advance way. If so, shouldn't it be the same for the tensor product? Is there a better way to approach this concept (and if so, why is it so often presented the formal/categorical way?), and where one should start?