If we go to the dual version, i.e. Wilson-Fisher without gauge field but just global symmetry, then U(1) symmetry simply corresponds to boson number conservation.
Now go back to Abelian-Higgs, you can ask what's the dual version of the previous global U(1) symmetry: you would then notice that the role of previous U(1) symmetry is played by the flux conservation now. This means your U(1) gauge field is non-compact, and monopoles operators are forbidden (at least in this simple duality of continuous theory).
Just remember here monopole operator is dual to previous boson operator, then other things are clear.