Omiyage is the Japanese social obligation of gifting. It's expected for travelers/guests to bring omiyage when they go somewhere for hosts or when they return home for family, friends or coworkers. It's best when the gift is a specialty product of the region, from which that the traveler is coming. Of course, in a country that values beauty in every detail, the omiyage has to be attractive and gift wrapped beautifully.
If you tell the shop attendant it's a gift, you'll receive a second paper bag, because they expect a bit of wear and tear on the first bag while you travel. The second paper bag will be fresh and crisp when you give the omiyage to its recipient.
As I had a plan to visit my friend in Kansai area today, I bought her some present yesterday. Attached is how the shop staff wrapped the gift for me. She offered me a second paper bag as well as wrapped the gift bag with another plastic bag because it was raining yesterday. This is how Japanese culture values beauty in every detail. Usually I don't like too much wrapping at stores and I make sure to bring my own shopping bags ("Eno-bag" in Japanese) for regular shopping. However, for special presents, I like it when the shop attendant take care of it with delicate manner!