Theme import not changing the color scheme
You have to export the color configuration from the old site and import it to the new site.

You can do this in UI. Go to /admin/config/development/configuration/single/export, select configuration type Simple configuration and the configuration name color.theme.mytheme. Copy and paste the color configuration to the new site. Use here the single item import section.

If you want to do this automatically when installing the theme you can place the file color.theme.mytheme.yml in the /config/install folder of the theme.