Topologically ordered superfluids (TOSFs) support deconfined, gapped anyonic excitations while remaining gapless because of Goldstone modes. We develop a general framework for characterizing their topological data in (2+1)-dimensional bosonic systems with spontaneously broken charge rm U(1)_(rm c) symmetry. Gauging rm U(1)_(rm c) maps a TOSF to a gapped topological order enriched by a dual rm U(1)_(rm dual) symmetry with vanishing Hall conductance, while gauging rm U(1)_(rm dual) recovers the TOSF. This two-way mapping allows the topological data of the TOSF, including the condensate charge, deconfined anyon content, and topological properties of vortices, to be systematically encoded using the algebraic framework of symmetry-enriched topological orders. In particular, superfluid vortices share the topological properties of flux defects associated with the residual discrete symmetry in the gapped anyon sector. We develop complementary algebraic and field-theoretic formulations of this framework, with the latter also suggesting a natural transition to a gapped topologically ordered phase. Additionally, using parton constructions, we present examples of TOSFs with non-Abelian vortices in bosonic systems.