Setting a label color of zoom extent in QGIS

It sounds you talk about mini map or overviews you could add in an atlas.

Two solution :

1. By styling your overview style and choose centroid symbol then your symbol will be font symbol then the text shown is based on data which will be a field of your atlas. (it will make an overlay)
2. By styling the atlas layer in your map shown in the picture you share. You can set the font to be based on rules and choose one in red if @atlas_featureid = $id and a black label that is based on the opposite rule @atlas_featureid != $id.

PS : if the layer you want to modify is not the one used for atlas let me know more about to adapt a rule for it.