  Some gauge theories with a spontaneously broken $U(1)$ symmetry exhibit
fractional Aharonov-Bohm (AB) phases around vortices in the Higgs regime. We
discuss continuity between confining and Higgs regimes in such gauge theories
with fundamental matter fields, focusing on the AB phases. By explicit
calculations in relevant lattice models, we demonstrate that the AB phase is
smoothly connected between the confining and Higgs regimes, supporting the
Higgs-confinement continuity. This result provides new insight into phase
structures of gauge theories with superfluidity, such as dense QCD.
