I apologize for answering my own question, but I think that the answer might be of some interest to others.
Since I was getting no answers I asked Professor Jacquet directly today. He explained that the expression for $L(St_G \times St_G,s)$ in the book cited above is not correct. The correct $L$-function appears in Proposition 1.4 of the paper A relation between automorphic forms of GL(2) and GL(3), joint with S. Gelbart. With the expression obtained in that paper we do have the expected equality $L(St_G \times St_G,s)=L(\sigma \otimes \sigma,s)$.