Mahler series with multiplicative coefficient sequences

We prove that every Mahler series, over a field of characteristic $0$, with multiplicative coefficients is regular in the sense of Allouche and Shallit. We also obtain an explicit characterization of such series. This yields a joint extension of the characterization of rational series with multiplicative coefficients (by Bézivin and Bell–Bruin–Coons) and of multiplicative automatic sequences (by Konieczny–Lemańczyk–Müllner). Both of these results are used in our characterization, so we do not obtain new proofs of these special cases.