If your data is the same type (polygons to polygons, etc) and is going to be styled according to the same fields you can save the symbology to a layer file, add it to your new maps, and change the source data to point to the new shapefiles.
If you need symbology based on different fields and just want a color match, you can use Export Map Styles to make all your symbols easily retrievable, they can be chosen from the standard symbols list that comes up when you go to choose a symbol. (This is also for if the data is the same type.)
If you have lots of colors on data that is of different types, you may be able to script something to export the RGB values of the symbols. I know I've seen scripts to color features based on RGB values in the table so maybe it can be made to go the other way.