The abstract for Shitov's paper is also a gem of brevity, here reproduced in its entirety:
"The chromatic number of G×H can be smaller than the minimum of the chromatic numbers of finite simple graphs G and H."
(And, of course, the first reference in the paper is co-authored by Erdős.)
The second example was used to show how colouring the tensor product may actually be useful.
I thought the second example was a little more complex (primarily because it requires the connection of incompatible nodes, as opposed to the more straightforward pairing of compatible nodes), so having a different, more straight forward example to just explain tensor products was helpful.
Also, having 2 different unrelated examples for the same thing can definitely help with understanding the underlying property/theorem.
And speaking as a graph theorist. When you've got three graphs like this, and you want to unambiguously refer to nodes of each, leaving things perfectly general can get quite tedious and torture the language. For example, sometimes it's just nicer to just say "red nodes" and "blue nodes" where any colors would suffice.