Hum! In 2.81 very strange behavior, You can select whole face but can individual vertex...
What ever method you are using for selection (box select, circle select or lasso select) if you select all vertex of the face then it's will be shown as selection otherwise not.
like whole face can be selected (in vertex selection mode):
In-short when all vertices of face is selected selection shows..
Note: blender 2.80 works fine