Is it possible to draw-create a picture even by hand, the diagonal union concept in set theory ? It is denoted by $\triangledown\cal A$ and defined to be $\{z ∈ Z : \text{for some }x ∈ z, z ∈ A_x\}$ where ${\cal A } = 〈A_x : x ∈ X〉$ is a sequence of subsets of $Z ⊆{\cal P }(X)$
There are various equivalent definitions of this, but none of which is intuitive enough.
The definition you give in your question is for taking the diagonal union of a family of sets, the elements of each of which are subsets of some fixed set $X$. A different definition that I think is more conducive being represented visually is for taking the diagonal union of a family of sets of ordinals, each of which is a subset of some fixed cardinal $\kappa$. On the surface, this other definition seems less general, but, since the Axiom of Choice implies that any set can be well-ordered, any diagonal union under your definition is can be regarded as one under this other definition.
For reference, here are the definitions found in Jech's Set Theory: The Third Millenium Edition (with notation changed to match that in your question):
Let $\langle X_\alpha : \alpha < \kappa \rangle$ be a sequence of subsets of $\kappa$. The diagonal intersection of $X_\alpha$ is
$$\underset{\alpha < \kappa}{\triangle} X_\alpha = \{ \xi < \kappa : \xi \in \bigcap_{\alpha < \xi} X_\alpha\}$$
and the diagonal union of $X_\alpha$ is
$$\underset{\alpha < \kappa}{\bigtriangledown} X_\alpha = \{ \xi < \kappa : \xi \in \bigcup_{\alpha < \xi} X_\alpha\}.$$
Each $X_\alpha$ is a subset of $\kappa$, so we can represent it visually as a tower of $\kappa$ many dots, with dots filled in for elements of $\kappa$ that are in $X_\alpha$, and dots left open for elements of $\kappa$ that are not in $X_\alpha$. For example, a typical $X_\alpha$ might look like this:
The entire sequence $\langle X_\alpha : \alpha < \kappa \rangle$ might look like this:
The "diagonal" that is relevant in the diagonal intersection and union is the following one:
For a given $\xi < \kappa$, we have $\xi \in \underset{\alpha < \kappa}{\triangle} X_\alpha$ if and only if every dot in the $\xi$th row to the left of the red line is filled in, and $\xi \in \underset{\alpha < \kappa}{\bigtriangledown} X_\alpha$ if and only if at least one dot in the $\xi$th row to the left of the red line is filled in. For example, in the picture I drew, we have $3 \in \underset{\alpha < \kappa}{\triangle} X_\alpha$ and $2 \in \underset{\alpha < \kappa}{\bigtriangledown} X_\alpha$.