No it cannot happen.

And not only for strongly $\mathbb{Z}$-graded rings; this is always the case for any strongly $G$-graded ring, where $G$ is a group. $A_k \otimes_{A_0} A_l \simeq A_{k+l}$ is an isomorphism of $A_0$-bimodules.

No it cannot happen.

And not only for strongly $\mathbb{Z}$-graded rings; this is always the case for any strongly $G$-graded ring, where $G$ is a group. $A_k \otimes_{A_0} A_l \simeq A_{k+l}$ is an isomorphism of $A_0$-bimodules.

(See: Corollary 3.1.2, p.82, from Methods of Graded Rings).