Whatever else one may or may not say about C2, if it is described with a single closed-shell determinant wave function (i.e., with Hartree-Fock or Kohn-Sham), one can rotate its occupied orbitals into two equivalent doubly-occupied non-standard sigma bond orbitals and two standard doubly occupied pi bond orbitals via orbital localization (Pipek-Mezey-like).
As the theoretical basis for bond order concept is closely related to this form of MO theory (which, for C2, is not entirely beyond question--it has low lying excited states), one could formally interpret this as a quadruple bond. Or as a double bond or triple bond, depending on how one wishes to interpret the non-standard sigma system.
In short, it is the perfect fighting ground for bored theorists 8).