The scale (conformal) anomaly can generate an electric current near the boundary of a system in the presence of a static magnetic field. The magnitude of this magnetization current, produced at zero temperature and in the absence of matter, is proportional to a beta function associated with the renormalization of the electric charge. Using first-principle lattice simulations, we investigate how the breaking of the scale symmetry affects this ``scale magnetic effect'' near a Dirichlet boundary in scalar QED (Abelian Higgs model). We demonstrate the interplay of the generated current with vortex excitations both in symmetric (normal) and broken (superconducting) phases and compare the results with the anomalous current produced in the conformal, scale-invariant regime. Possible experimental signatures of the effect in Dirac semimetals are discussed.