Please see this following answer to a question. https://math.stackexchange.com/a/2384534/343701
Two requests: 1) Am I correct, and 2) There is another question at the end of the answer, which is the subject of this question. Could please help me understand that concept?
I'll be honest: I'm not entirely sure what you're saying in the body of your answer. What confuses me is when you write $V(1 \to A) = 1 \to B$. To me, $1 \to A$ is a signature, meaning that it just tells me what the domain and codomain of a function are. So when you put $V(\ldots)$ around it, I'm not sure what to think. But I'm going to try and write out what I think you were saying, and try to answer your question as well.
For Lawvere, a variable element is a map $e : D \to X$; $e$ is the element, and $D$ is the "domain of variation". Given any particular element $d : 1 \to D$ in the domain (where $1$ is a constant domain of variation, so that $d$ doesn't vary at all), we can compose to get $e \circ d : 1 \to X$, which is a constant element of $X$.
For example, if think of the "the temperature" as a variable number depending on where we put our thermometer, then we would write $t : P \to N$ where $P$ represents the places we could put our thermometer and $N$ represents our notion of numbers. Then, given any particular place $p : 1 \to P$ we put our thermometer, we get a particular number $t \circ p : 1 \to N$ which we can read off the thermometer.
Now for more general figures. Suppose that we don't just want to know what the termperature is at one place, but we want to know how the temperature changes as we take a walk. Suppose that we intend to walk in a loop, so that our walk will trace out the shape $L$ of a loop in the places $P$. In other words, our walk should be a map $w : L \to P$, a "figure whose shape is the loop ($L$) in the space of places ($P$)". Then, we can watch the thermometer as we walk to see the temperature $t \circ w : L \to N$ as it varies around the loop.
This works particularly well if we time our walk (so that everything is nice and precise). Suppose we bring a pocketwatch, which is a disk with evenly spaced numbers on its perimeter. Lets suppose that we will walk for exactly an hour, which is represented by exactly one rotation of the minute hand around the watch face. Therefore, our loop $L$ is now precisely determined as the perimeter of the pocketwatch, and we see that the walk $w : L \to P$ assigns each "time" (that is, position of the minute hand on the loop) to a place in $P$. Then, if we watch our watch while making note of the temperature, we get a variable number $t \circ w : L \to N$. For any particular time (say, at $20 \text{ minutes} : 1 \to L$), we get a particular a particular temperature ($t \circ w \circ (20 \text{ minutes})$).
So figures, like variable elements, are just maps. What makes them "figures" rather than "variable elements" is how we use them and how we think about them. For example, we are more likely to call something a "figure" when the domain is particularly simple (like a loop), and when we care about its whole shape rather than the behavior of a varying part. But as the above example shows, these two ideas can blur together; we might have thought of the walk as having the shape of a loop in our space of places (therefore thinking of $w : L \to P$ as a "figure"), but we then thought of the temperature as we went on the walk as a number that varied as we walked (therefore as a "variable element" $t \circ w : L \to N$). The domain in both these cases is the loop $L$, but the first we thought about as a figure, and the second as a variable element.
Now to your question. If we agree with Lawvere that maps $f : S \to E$ can be thought of as figures $f$ of shape $S$ in some environment $E$, then in particular the map $\textbf{id}_S : S \to S$ should be a figure $\textbf{id}_S$ of shape $S$ in the environment $S$. What Lawvere is saying in your quote is that we can read $\textbf{id}_S$ in two ways:
We call $\textbf{id}_S$ the generic figure of $S$ because every other figure of shape $S$ is the image of it under composition (can you see why?). We call it an independent variable because substituting it into anything (that is, precomposing) causes no changes.
So, in the case where we think of $L$ as being "a loop", so that a map $\ell : L \to X$ is "a figure in $X$ whose shape is a loop", or for short: "A loop $\ell$ in $X$''. But by this way of speaking, $\textbf{id}_L$ is "A loop $\textbf{id}_L$ in $L$". This makes sense: of course there is a loop in a loop!
But as we saw above, the loop $L$ might be the marked perimeter of a pocketwatch whose constant elements are therefore the possible positions of the minute hand. In this case, a map $\ell : L \to X$ might be better thought of as "an element $\ell$ of $X$ which varies with the minute hand of a pocketwatch". Then the map $\textbf{id}_L$ should be thought of as "the position of the minute hand varying with the position of the minute hand", an idependently varying element in the sense that it depends on nothing but itself!
I hope this was helpful and that I haven't confused things further.
Cheers,