The various real life applications of graph products are huge, a few of which I hope to be able to successfully describe are as follows:
$1.$ Graphs arising in chemistry are a primary source of examples for graph theory: chemical trees, and fullerenes are just a few of the examples. After a molecule is represented as a graph, the primary goal of chemical graph theory is to investigate the graph and to predict the molecule’s properties by computing carefully selected graph invariants. The Wiener index is the oldest such invariant.
$2.$ Another application includes a graph invariant called windex, introduced by Chung, Graham, and Saks in the context of dynamic location theory. It is closely connected to Cartesian products of complete graphs. These graphs are also known as known as Hamming graphs.
$3.$ Networks arise in many different areas, such as mathematical chemistry, software technology, and operations research. And, the investigation of very complex graphs and networks became an important research topic in the last decade, coinciding with increased interest in the Internet, citation networks, and neural networks. To model large networks, Kronecker Graphs are used.
You can also read this link for more information on the applications of graph products. Hope it helps.