I am using a Parent theme to set up most of my customizer settings and Parent theme. Then creating a child theme that has additional controls/settings and different layout. Does the customizer data move its values from theme to theme if they have the same value name or does it need to set values again for each theme?
The Customizer, by default, uses the Theme Mods API, which saves settings as Theme-dependent – meaning that parent and child Themes will have a unique option in wp_options. The Customizer can be configured to use the Options API, and as long as both parent and child Themes reference the same option name, settings can be shared.