If we have a set of linear symmetry currents $J^{\mu}_{\alpha}$ and attempt to find if they are anomalous, we find that if we change the regularization procedure, the anomaly will get mixed around the currents. In particular, for ordinary massless QED, the canonical form of the anomaly is purely in the chiral symmetry. But it is possible to do the integral such that the gauge symmetry is also anomalous.
Why is this an acceptable fact? In the case of QED, there is the regularization independent fact that there is an anomaly, but where the anomaly resides is regularization dependent. So what does this mean? A different way of calculating the anomaly produces a different theory?
My understanding is the fundamental idea of the renormalization group is that observable quantities cannot depend on the method of regularization. Anomalies are observable, otherwise it could not have been used to explain the neutral pion decaying to two photons amplitude.
