Altermagnetism is a recently identified magnetic phase with zero net magnetization but momentum-dependent spin splitting, for which unambiguous experimental signatures remain scarce. We show that thin superconducting films coexisting with collinear $d$-wave altermagnetic order exhibit characteristic fourfold anisotropies in their critical temperature, in-plane critical field, and critical current density. We demonstrate that these anisotropies originate from the interplay between a paramagnetic-type coupling — generated by an exchange or Zeeman fields — and the orbital coupling also induced by the magnetic field. The altermagnetic contributions to the predicted anisotropies are odd in the applied magnetic field. Observed together with zero net magnetization, these signatures directly indicate the presence of altermagnetic order and allow it to be distinguished from anisotropies of other origins. Our results, applicable to intrinsic altermagnetic superconductors and to superconductor/altermagnetic-insulator heterostructures, provide directly measurable signatures for identifying altermagnetic order in thin-film geometries.