For some reason, pluralization of singular brands is a thing depending on your dialect of English. I won’t state the regions because I don’t wanna accidentally narrow down where someone lives. Although I guess referencing Aldi already does that a bit.
You’re definitely right about the possessive thing. I’m not sure about the snow clone. I always thought of it like they were treating the name of the store like the last name of the family who owns it. So it’s like a shortening of, “we’re going down to the Aldi’s store.” As if you’d walk in and Mr. Aldi would be behind the counter.
Others, I'm sure. Where do Australians go? Bottle-o always sounded very sing song in a Tolkien sort of way. There's a kebab place we call "Ararat's", but I don't remember the official name. I sometimes remove the possessive/plural to sound funny (bunning)