Use Select by expression with this expression and replace region_layer with your layer name:

within ( $geometry, overlay_nearest ('region_layer', buffer($geometry,0.1))[0] ) 

The expression looks for each commune if it is completely within the nearest feature. For exact matches add a very small buffer around the region. This does not affect neighboring communes being selected, as we only select if a commune is completely within the region.

Screenshot: only those of the red polygons are selected (highlighted in yellow) that are completely within the blue polygon:

Use Select by expression with this expression and replace region_layer with your layer name:

within ( $geometry, overlay_nearest ('region_layer', buffer($geometry,0.1))[0] ) 

The expression looks for each commune if it is completely within the nearest feature from the region_layer. For exact matches add a very small buffer around the region. This does not affect neighboring communes being selected, as we only select if a commune is completely within the region (not not just overlaps in the margins).

Screenshot: only those of the red polygons are selected (highlighted in yellow) that are completely within the blue polygon: