Hei, my bag is also from this brand! It's what I called in a former post a foldable tote bag.
In many European countries supermarkets sell reusabke shopping bags made out of recycled PET bottles for a few bucks, certainly under 5€. They are bigger and more durable than the paper bag and hold also heavy items. They are not water proof but don't break when wet. Sometimes they have a nice print and often it's only a form of advertising for the shop. I usually take one of these with me when travelling and I first thought to use it for the pool. But then I would have to bring another bag when just wandering from one place to another on the ship.
The shopping bag can be used in a multitude of ways too. You already listed a few things. I use the supermarket shopping bags also for recycling paper, PET and glass at home, bringing food and gifts to parties or to transport laundry to the laundromat while traveling.