Catalysis of the hydrolysis of an isopeptide bond. An isopeptide bond is an amide linkage between a carboxyl group of one amino acid and an amino group of another amino acid in which at least one of these groups is not on the a-carbon of one of the amino acids (for example, the link between an epsilon-amino group of a lysine molecule to a carboxyl group on a second amino acid is an isopeptide bond).