This answer is the synthesis of the two comments, @Erik and mine.
Shapefile is a GIS file format. Shapefiles are a format invented and controlled by Esri (the term proprietary is discutable, see the comments of this answer), but with a large open specifications for interoperability (GIS & CAD software mainly).
Following the documentation, this is the same format, the only "small" difference can be in the sidecar files, QGIS can have "qix" for spatial index and "qpj" for projections, see here.
The only thing not compatible between ArcGIS and QGIS are the style files which come with some shapefiles :
*.style for ArcGIS and
*.sld for QGIS.
Over the last few years the open source community tried to push the geopackage file format, which solves some issues which Shapefile format suffer from.