The bivariate $P$- and $Q$-polynomial structures of association schemes based on attenuated spaces are examined using recurrence and difference relations of the bivariate polynomials which form the eigenvalues of the scheme. These bispectral properties are obtained from contiguity relations of univariate dual $q$-Hahn and affine $q$-Krawtchouk polynomials. The bispectral algebra associated to the bivariate polynomials is investigated, as well as the subconstituent algebra of the schemes. The properties of the schemes are compared to those of the non-binary Johnson schemes through a limit.