Yes, there are similarity relations where one can describe a common factor that underlies the similarity. A mother and a daughter may share same eye colors, nose shapes, etc (the analogy in the bbc episode seems indeed faulty). Wittgenstein's claim seems to be that the picturing relation is just not one of these relations. There is no describable common factor to that similarly.
To inquire why this might be, let's take an example. How does the sentence
the cat is on the mat
picture the state of affairs where the cat is on the mat? What is the common structure between the sentence and the SOA? Well, it seems to be something like:
In both instances there are two objects, and a certain relation between them
The two objects being, on the one hand the cat and the mat, on the other hand the expressions "the cat" and "the mat".
We have just described the common structure, haven't we? It is just that the terms that we had to use in the description are "senseless" by Wittgenstein's lights.
6.54 My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.)
We have spoken about two objects. But there is really no way to count objects, as such. "Object" is not a true concept, but a pseudo-concept, a place-holder. As Bertrand Russell put it, in his Introduction to the Tractatus:
Wittgenstein contends and, I think, rightly . . . that “object” is a pseudo-concept. To say “x is an object” is to say nothing. It follows from this that we cannot make such statements as “there are more than three objects in the world,” or “there are an infinite number of objects in the world.” Objects can only be mentioned in connexion with some definite property.
So the only way to describe the picturing relation is in "Tractarian" senseless terms; using pseudo-concepts such as "object". This does not amount, by Wittgenstein's lights, to saying anything. It only amounts to showing something.