TWebColorPicker

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.
https://developer.mozilla.org/en-US/docs/Web/HTML/Element/input/color
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.