Two-dimensional magnets have manifested themselves as promising candidates for quantum devices. We here report that the edge and strain effects during the device fabrication with two-dimensional honeycomb ferromagnets such as CrX (X=Cl, I, Br) and CrXTe (X=Si, Ge) can be characterized by a (1+1)-dimensional magnon chiral anomaly beyond the fermionic paradigm. In the presence of zigzag edges, a pair of chiral bulk-edge magnon bands appear and cause an imbalance of left- and right-chirality magnons when subjected to nonuniform temperature or magnetic fields. In the presence of a uniaxial strain, the bulk Dirac magnons are broken into chiral magnon pseudo-Landau levels, resulting in a magnon chiral anomaly observable through a negative strain-resistivity of the magnetic dipole and heat. Our work demonstrates a chiral anomaly with (quasi)particles obeying non-fermionic statistics and will be instructive in understanding anomalous magnon transport.