Was there a reason not to offer the system or predefined colors? Its somehow more difficult when you don't know the color RGB values

Do you mean at design-time or at runtime?

During runtime

The control wraps the HTML5 standard color picker.
How this control looks and what colors it offers depends on browser and operating system.
If you would want control over colors displayed, a custom color picker should be used. One such color picker you can use in TMS WEB Core applications is TTMSFNCColorPicker.

I thought the controls available in VSC Web Core extension are a subset of the TMS Web Core for Delphi/Lazarus. So TTMSFNCColorPicker might be available in one of the upcoming versions of the VSC Web Core extension?

Installable add-on components (such as FNC components) will be supported in the next version of TMS WEB Core for Visual Studio Code.
You could use & create additional controls such as FNC at runtime though.