Why is the diagonal of the infinitesimal square contained in $D(2)$? The object $D(2)$ is the first order infinitesimal neighbourhood of the plane in synthetic differential geometry. For example in the algebraic geometric model over the base ring $\mathbb C$, the space $D(2)$ is the subscheme $\operatorname{Spec}\mathbb C[x,y]/(x,y)^2$ of the plane. People draw $D(2)$ as a tiny circle around the origin. The shape makes sense, since for example $D(2)$ is invariant under $SO(2)$-transformations of the plane.
The infinitesimal object $D\times D$ is a tiny square around the origin, and since $D\times D = \{(x,y):\mathbb A^2|x^2=0\wedge y^2=0\}$ we see that $D(2)$ is contained in $D^2$. The picture is that of a tiny circle inside a tiny square. But here is something that confuses me.
The diagonal of the tiny square $D^2$ in the plane is the subspace of the form $\Delta_{D^2} =\{(\delta,\delta):\mathbb A^2|\delta^2=0\}$. From the geometric picture the diagonal of $D^2$ should not be contained in $D(2)$, but it is. We have that $\Delta_{D^2}\subset D(2)$ since combinations of the two entries of $(\delta,\delta)$ always multiply to zero when $\delta^2=0$. What is wrong with the geometric picture? Is there a better way to picture infinitesimals? I have drawn a picture below.
 A: tl;dr There are two issues with your geometric reasoning. First, drawing $D(2)$ as a tiny circle around the origin is not really accurate. Second, your argument about squares and circles is not the sort that works in synthetic differential geometry, because it cannot handle degenerate cases. There are other ways of picturing infinitesimals, such as the one below.
I. Not visualizing the smooth real line
Many things go wrong when one tries to visualize the infinitesimal neighborhood $D=\{d \in \mathcal{R}\mid d^2=0 \}$ as a scaled-down copy of the smooth real line $\mathcal{R}$. The picture is not just intuitively misleading: its formalized versions are false. For example, a scaled-down copy $\varepsilon \mathcal{R}$ remains closed under addition, while $D$ being closed under addition would contradict the Kock-Lawvere axiom. 
Things get worse once we add order/integration axioms into the mix! For arbitrary $d_1, d_2 \in D$, both $d_1 \leq d_2$ and $d_2 \leq d_1$ necessarily hold: we can't really speak of an infinitesimal lying to the right/left of another.
It follows that all geometric pictures of infinitesimal geometry will be quite misleading. E.g. when you draw an infinitesimal rectangle around the origin, you don't really know its orientation, and you can't even rule out its degeneracy (i.e. that one or more vertices coincide) in any way.
We don't even need to invoke 2-dimensional geometry to see this. When we write "consider an infinitesimal closed interval of length $d \in D$", for some arguments we might want to rule out the empty interval $\{0\}$. But we can't just assume $d \neq 0$, as there is (provably) no element $d \in D$ that satisfies $d \neq 0$.
Things might look extraordinarily bad at this point: one of the purported benefits of SDG is that it's supposed to make certain geometric arguments in Lie theory perfectly rigorous. But now it seems that even ordinary geometric reasoning breaks down!
You can read more about this phenomenon (and why it's not as bad as it looks) in my answer to another question. While you can prove that the infinitesimal neighborhood $D$ of $0$ does not coincide with $\{0\}$, you can also prove that no point $d \in D$ is really distinct from $0$ in the sense that $\neg d \neq 0$ holds for all $d \in D$. This all works out thanks to intuitionistic logic. It allows $D$ to behave like an interval in certain aspects (thus the Kock-Lawvere axiom can be consistently adopted). But $D$ will still behave very much like a single point in other aspects. The same goes for the geometric objects built using $D$: in some of their aspects they'll behave like ordinary rectangles or circles, but in many other aspects they'll act as if they were single points.
Thus, most geometric arguments about such shapes break down in the infinitesimal world. Only certain arguments keep working, ones that can handle potentially degenerate objects gracefully.
II. What is wrong with your geometric picture above?
There are two issues. The first issue is that drawing $D(2)$ as a disk (and to a lesser extend $D^2$ as a filled square) is at best a visual aid to remember the inclusion $D(2) \subseteq D^2$, and not really an accurate picture. As discussed above, these sets are not infinitesimally scaled versions of disks and rectangles at all.
The second issue is that not all geometric arguments remain valid in the infinitesimal world. Only those arguments keep working that can handle degenerate objects gracefully. But your argument ("from the geometric picture the diagonal of the square should not be contained in the circle") does not handle degenerate objects gracefully! In particular, the degenerate diagonal of a degenerate square (of side length $0$, the point $\{0\}^2 \subseteq \mathcal{R}^2$) is contained inside the degenerate circle (of radius $0$) around the origin.
III. Visualizing the smooth real line
Long story short, I recommend not conceptualizing the object $D$ as a tiny version of the line $\mathcal{R}$. This leads us to your second question: is there a better way to picture infinitesimals?
As I explained in an answer to a similar question I usually think of an infinitely long chain built from tiny metallic links. From afar, you see a line. From up close, you see chain links labeled by ordinary numbers: but the links themselves have richer, more exotic geometric structure. The first-order infinitesimal neighborhoods of numbers lie on these links.
When we apply a function to the real line, we twist and turn this chain into a seemingly arbitrarily complicated curve. However, when we focus on a single link, we see that the transformation behaves linearly. The transformation preserves the integrity of the links and the overall shape of each link remains the same, even though its orientation may have changed. This is the Kock-Lawvere axiom.

The picture somewhat extends to higher dimensions: you can visualize the plane $\mathcal{R}^2$ as a sort of chain mail, where neighborhoods of the origin such as $D(2)$ and $D^2$ live inside a single ring piece, which has more complicated structure than the large-scale structure of the plane. But in the end, this visualization is still more of a "useful reminder" than something deep that could be made rigorous or formalized.
