Breaking single polygon into multiple polygons?

I think that what you actually want to do is to convert multipart polygons into singlepart polygons.

In ArcGIS it can be done using Multipart To Singlepart (Data Management) tool.

In QGIS vector menu, as well as in the Processing Toolbox, you'll find Multipart To Singlepart.

This algorithm takes a vector layer with multipart geometries and generates a new one in which all geometries contain a single part. Features with multipart geometries are divided in as many different features as parts the geometry contain, and the same attributes are used for each of them.

You should specify option SINGLE_PART when running ArcGIS Dissolve tool:

SINGLE_PART —Specifies multipart features are not allowed. Instead of creating multipart features, individual features will be created for each part.