I use the Topology Checker Plugin in QGIS 3 with the rule "must not overlap" for a single PostGIS multipolygon layer.
Can somebody explain the meaning of errors (colored in red) when the Plugin colors a whole polygon as a red area?
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When the topology checker highlights a feature in red, it means that feature violates one of the rules.
Since your only rule is "must not overlap," when a polygon is red it means that polygon overlaps with another polygon.
It's up to you to find the exact place where it overlaps. I suggest using the Clip tool (from the Vector menu > Geoprocessing tools) to find any overlaps. The clip tool will create a new layer that is just the areas where the polygons overlap. You can zoom to each individual area from the attribute table (select one feature, click the "zoom to selection" button). Then fix the overlaps in the original layer.