Imagine if suddenly every grocery, pharmacy, petrol station, parking place, restaurant, bar etc. now would ask you for your ID AND would snap a picture and store in their database - you wouldn't be happy about it.
But you do have a point about "storing the picture". I think that's why it's very important for whatever solution is chosen to be something that proves you're old enough without saying who you are.
At least in my country, the ID app lets you generate 3 levels of QR:
Level 1: Just age (also shows a photo on the screen). This is what you would typically use to go in a club or buy alcohol.
Level 2: Adds Full name, birth date, validity date.
Level 3: All the data you can see on the physical ID card.
As for why would they, the same reason there are hundreds of tracking cookies on every site.
Oh, wait...
The big companies are still mining user data, they are just forced to use some extra dark patterns to trick people into compliance. Would-be criminals are not going to stop being criminals because of the threat of fines. And TLAs are not going to wait for due process to acquire access to data legally.
All that GDPR does is give the illusion that people are being protected and CYA for politicians and bureaucrats when asked "what are you doing about evil Zuckerberg?"
I don't want to have to read your noise.
Unless you live at some place where they still accept cash of course, but the writing is on the wall already.