Bases are different .
Having the paint made by another co. would not produce the same color.
Better to color match.

They all have a database of one kind or another with competitor's color palettes and formulas.

You'll get one of two things when you cross brands:

1. The paint counter person will look up your color and sell you the closest match in their home-store brand. Depending on the brand, they have a set point, Delta E minimum or some kind of corporate standard for a color match. Universal Delta E scale is 0-1 not detectable by human eye, 1-2 narrow difference noticeable to a trained eye only, 2-3.5+ Obvious to just about any human with eyeballs and normal vision.

If the computer color match from their brand palette is less than two (ish), they may mix and sell you that color. Because it's close enough per Delta E standards to the other brand's color -- and odds are you'll never know the difference.

2. They look up your color, find a match in their brand and use that data to START making you a custom color. A series of mixing, draw-downs and evaluation will follow until you leave the store happy that you got the color your came for.

YMMV. Meaning there's probably other options too. All paint stores are different. Communication is important. Talk to the counter person so they can explain how they do it in their store.