Little known fact, Photoshop has two distinct versions of colour modes.
One is RGB vs CMYK etc. You're probably familiar with this and the problems/issues of incompatibility between some of these colours.
But there's another thing entirely, called colour spaces. It's annoying as all heck to learn, and of little benefit to the normal user... and since iOS and Android design has become a thing, it's actually better to turn it off, completely.
Firstly, when/if creating a new document, you can set the colour settings here:
And once you're inside a document, you can fiddle with this wondrous myriad of nonsense:
If you get both of these to match, plus sort out that you're using the same color mode (16 bit and RGB vs CMYK etc) in both documents, you should be able to get the same colours in both documents.
Pick one of the documents, and match the other one's settings to it, I propose.