Our big home improvement stores don't have house name brand, so when there is a paint mixing station it's still one from a paint brand.
That paint brand will most likely also have their product on shelf, along with several other brands.
Quite frankly when I had to chose a new color for my bedroom I only looked at one brands catalogue (I only made sure it was the same price range as the other brands) or you are quickly overwhelmed, and picked up a 5/8 of a gallon off the shelf, same for the primer.
They can probably order it with you want 'pastel lilac' from brandX and they do carry brandX, they just don't have 'pastel lilac' in stock.
Even our fairly small, non-chain hardware store mixes paint to order. That's just a given here.
It's a lot less shelf space devoted to paint when they're mixed to order. Consider: if you have just 10 pre-mixed colors, times the four common finishes (flat, eggshell, satin, and semi-gloss) times gallons and pints (OK, the metric people can substitutes liters for the pint cans -- do y'all have larger cans, hold about 4 liters?) that's 80 cans of paint that you have to find room for. And I see that I forgot that you need both interior and exterior paint, so double that. 160 cans of paint. That's if your store only carries one brand of paint. Usually they have at least two.
Mixed to order: light base, medium base, deep base x 4 finishes x pints/gallon x exterior/interior = 48 cans of paint, not 160.
What's cool is that in the last couple of years, they've started selling little sample jars for about $3, also mixed to order. You take it home, slap it on the wall, and you can see just how BRIGHT
that blue is, which you can't really tell from the little sample chip. (Why, no our house wasn't a landmark BLUE
house, why do you ask?)