Internally, Pokemon have a primary color value assigned to them. For example, Bulbasaur has a primary color of Green. This value can't be seen, so far as I know, in Pokemon Sword/Shield or Pokemon Scarlet/Violet. However, it can be seen in Pokemon Home by filtering in the Pokedex.
Here's the catch: color data in Pokemon Scarlet/Violet or Sword/Shield is not always identical to Pokemon Home. For example, Dialga-Origin is classified internally as White in Scarlet/Violet, but as Blue in Home (see above screenshot). Same thing applies to Sneasler, who is Purple in Scarlet/Violet but Blue in Home. Unless there is in fact a way to search for colors in SwSh or SV or a feature where color matters, I think it makes more sense to use the Home colors as opposed to the specific game's colors, since that's the only practical way it can be referenced.
I have not done a full comparison from Home vs the individual games, but comparing Home data vs Showdown data (which is hopefully at least mostly accurate) I found the following differences.
I have not done a full comparison from Home vs the individual games, but comparing Home data vs Showdown data (which is hopefully at least mostly accurate) I found the following differences.
Dialga-Origin: White -> Blue
Morpeko-Hangry: Purple -> Yellow
Zarude: Black -> Green
Zarude-Dada: Black -> Green
Wyrdeer: White -> Gray
Sneasler: Purple -> Blue
Overqwil: Gray -> Black
Ceruledge: Blue -> Purple
Tatsugiri-Curly: Pink -> Red
Brute Bonnet: Gray -> White
Slither Wing: Red -> White
Iron Moth: Yellow -> White
Arctibax: Gray -> Blue
Baxcalibur: Gray -> Blue
Gimmighoul: Brown -> Red
Gimmighoul-Roaming: Blue -> Gray
Roaring Moon: Green -> Blue
Miraidon: Blue -> Purple
Morpeko-Hangry: Purple -> Yellow
Zarude: Black -> Green
Zarude-Dada: Black -> Green
Wyrdeer: White -> Gray
Sneasler: Purple -> Blue
Overqwil: Gray -> Black
Ceruledge: Blue -> Purple
Tatsugiri-Curly: Pink -> Red
Brute Bonnet: Gray -> White
Slither Wing: Red -> White
Iron Moth: Yellow -> White
Arctibax: Gray -> Blue
Baxcalibur: Gray -> Blue
Gimmighoul: Brown -> Red
Gimmighoul-Roaming: Blue -> Gray
Roaring Moon: Green -> Blue
Miraidon: Blue -> Purple
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