There is a polygon layer with several polygons. They are displayed with some colour which can be changed simultaneously for all of them. Is there a way to manually select several polygons (without referring to their attributes in the table) and to pick for them some other colour (that is, by manually selecting polygons and groups of polygons one by one to assign them some different colours to display)?
Usually, the color is defined based on a field in the attribute table. It is often the best method on the long term (e.g. to reproduce what you did) even if you might have the feeling that you are wasting time at first. There are a few workaround :
if you just need one color, a) you can set a specific color for the selection in your feature class properties. Then when you select a polygon it takes the color that you chose. bà as already mentioned, you can create a new layer from the selection. In the first case, it will disappear when you unselect. In the other case, it will stay.
if you need more colors, convert you features to graphics. With graphic, you can change the color directly at the properties level.
Create Layer from Selected Features is likely the easiest way to go about this.
Select the features that you would like to change symbology
Right Click on that layer in Table of Contents
Selection-->Create Layer from Selected Features
From here, you can take the subset and symbolize how you see fit.