They are different products. PostGIS is a collection of spatial operators, tables and datatypes that overlays PostGreSQL, a relational database. These added functions and data constructs enable management of spatial data and querying using spatial data operators.
ArcMap is a user interface for interacting with spatial data. ArcMap is designed to operate on various spatial data types. The native database format is the ESRI geodatabase, which is essentially a relational database with a collection of special tables added for managing spatial data according to ESRI conventions. In current versions for ArcGIS, PostGIS can host a geodatabase. Geodatabases can also be instantiated using a file-based structure (file geodatabase), in Microsoft Access (personal geodatabase), or in MS SQL server or Oracle.