We consider multivariate stationary processes $(\boldsymbol{X}_t)$ satisfying a stochastic recurrence equation of the form $$ \boldsymbol{X}_t= \mathbb{ M}_t \boldsymbol{X}_{t-1} + \boldsymbol{Q}_t, $$ where $(\boldsymbol{Q}_t)$ are iid random vectors and $$ \mathbb{M}_t=\mathrm{Diag}(b_1+c_1 M_t, \dots, b_d+c_d M_t) $$ are iid diagonal matrices and $(M_t)$ are iid random variables. % It is known that under suitable assumptions the marginals $X_{t,i}$ of $\boldsymbol{X}_t$ are regularly varying. We obtain a full characterization of the Vector Scaling Regular Variation properties of $(\boldsymbol{X}_t)$, proving that some coordinates $X_{t,i}$ and $X_{t,j}$ are asymptotically independent even though all coordinates rely on the same random input $(M_t)$. We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK-ARCH and CCC-GARCH) processes and to log-returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.