So, for one thing, it's not possible for retailers because very few (probably none) store payment credentials anymore. Too expensive - that's what payment processors are for.
For payment processors themselves, clearly they can, it's the subject of the parent post, but there are regulatory hurdles.
This is specifically for the case where the retailer does not want to store these details. Heck they can even get away with not building a user account system at all. Just a store listings page and a few lines of code imported from Stripe.
I'll add that some countries now have rules that prohibit retailers from storing the payments information of customers.
I don't know the details, but India now has such rules in place.
For retailers to store customer information (for recurring payments and whatnot) — from what I understand — they cannot directly store the card details but instead have to generate a token or something and use that.
In India, big companies (like Amazon, Netflix etc) seems to have complied, but many smaller ones haven't.