Yes, that is a real flag. I do not know how many flags were brought with the astronauts to the Moon. Back in the Apollo days, the astronauts were allowed to bring 5 pounds of items with them to be carried aboard the CSM, e.g. 15 pounds in total. On Moon landing missions the astronauts could also place 1/2 pound each aboard the LM, e.g 1.5 pounds in total. These items first had to be OK'ed by Deke Slayton. They were then placed in bags, and placed aboard the respective spacecraft. These items were then called the Personal Preference Kit (PPK). There were also Official Preference Kits (OPK) that were placed aboard the spacecraft by NASA. I do not know the weight restrictions on the OPK. These OPKs had flags, patches etc. that NASA would then give out to assorted people, such as senators, congressmen, employees, contractors etc. etc. Within the PPKs the astronauts brought a wide variety of lightweight items, such as flags, patches, coins, stamps, jewelry etc. While in theory somewhere or other there should be listings of what was in PPK and OPK for the assorted flights, research by quite a few interested authors, collectors, historians etc. has not found these lists. One other point, many of the flags, again, I don't know what percentage, were either (US) State flags, or flags of other countries.
@Skyman , thank you so much for taking the time to write these up. Each submission was very interesting and I learned a lot about this historic mission. I look forward to the next mission in November!