We analyze the number of cliques of given size and the size of the largest clique in tensor product $G \times H$ of two Erdős-Rényi graphs $G$ and $H$. Then an extended clustering coefficient is introduced and is studied for $G \times H$. Restriction to the standard clustering coefficient has a direct relation to the local efficiency of the graph, and the results are also interpreted in terms of the efficiency. As a last statistic of interest, the number of isolated vertices is analyzed for $G \times H$. The paper is concluded with a discussion of the modular product of random graphs, and the relation to the maximum common subgraph problem.