Tensor product of linear codes and generators
Yes, the direct product code is denoted in this way and has this generator matrix (see MacWilliams, Sloane p.569). Some authors define $C_1 \otimes C_2$ to be the code with parity check matrix $H_1 \otimes H_2$. To see that the generator matrix definition is equivalent, $$(G_1 \otimes G_2)(H_1 \otimes H_2)^T = (G_1 \otimes G_2)(H_1^T \otimes H_2^T) = G_1 H_1^T \otimes G_2 H_2^T = 0 \otimes 0 = 0.$$ Likewise, $(H_1 \otimes H_2)(G_1 \otimes G_2)^T = 0$.