No: there is a long-standing tradition whereby winners get to keep captured banners.
A number of captured Civil War Flags have been returned south over the years. Sometimes as part of an exchange of artifacts between museums, sometimes as pure gifts. However, I doubt that this flag was that of Virginia battalion. Why? Because during the Civil War, a battalion was an ad hoc subunit of a
regiment given command of some of the companies thereof. Therefore it was likely a company or regimental flag, not a battalion flag.