There is a well-known result by Quillen stating that if $X_A$ is the Severi-Brauer variety of a central simple algebra $A$ of degree $d$ over a field $k$, then its (Quillen) K-theory decomposes as $$K_\bullet(X_A) \simeq \bigoplus_{i=0}^{d-1}K_\bullet(A^{\otimes i}).$$

In every text I've seen mentioning it, the decomposition is stated at the level of $K_n$, and is just taken as a decomposition of abelian groups. But since $X_A$ is a scheme, $K_\bullet(X_A)$ is a graded-commutative ring, and this induces a multiplicative structure on $\bigoplus_{i=0}^{d-1} K_\bullet(A^{\otimes i})$.

It seems almost compelling that this structure should come from $$K_p(A^{\otimes i})\otimes K_q(A^{\otimes j})\to K_{p+q}(A^{\otimes i+j})\to K_{p+q}(A^{\otimes r})$$ where $r\equiv i+j$ modulo $d$, and the last map is the isomorphism given by Morita equivalence.

Nonetheless, I couldn't see any reference to that fact in the literature I've read. Is it a "folklore" result? Is is actually stated/proved somewhere and I've missed it?

Actually, even in the split case, where we get $K_\bullet(\mathbb{P}^{d-1} _k)\simeq K_\bullet(k)^d$, I haven't seen stated that this gives an isomorphism with the group ring $K_\bullet(k)[\mathbb{Z}/d\mathbb{Z}]$.