We consider multivariate stationary processes $(\boldsymbol{X}_t)$ satisfying a stochastic recurrence equation of the form $$ \boldsymbol{X}_t= \boldsymbol{m}M_t \boldsymbol{X}_{t-1} + \boldsymbol{Q}_t,$$ where $(M_t)$ and $(\boldsymbol{Q}_t)$ are iid random variables and random vectors, respectively, and $\boldsymbol{m}=\mathrm{diag}(m_1, \dots, m_d)$ is a deterministic diagonal matrix. We obtain a full characterization of the multivariate regular variation properties of $(\boldsymbol{X}_t)$, proving that coordinates $X_{t,i}$ and $X_{t,j}$ are asymptotically independent if and only if $m_i \neq m_j$; even though all coordinates rely on the same random input $(M_t)$. We describe extremal properties of $(\boldsymbol{X}_t)$ in the framework of vector scaling regular variation. Our results are applied to multivariate autoregressive conditional heteroskedasticity (ARCH) processes.