Hydrolase that deubiquitinates polyubiquitinated target proteins such as MDM2, MDM4 and CCND1. Possesses both ubiquitin-specific peptidase and isopeptidase activities. May play a role in the regulation of the circadian clock. Thiol-dependent hydrolysis of ester, thioester, amide, peptide and isopeptide bonds formed by the C-terminal Gly of ubiquitin (a 76-residue protein attached to proteins as an intracellular targeting signal). Homooligomer. Belongs to the peptidase C19 family. USP2 subfamily.