Click the arrow next to the Font Color icon to activate a toolbar from which you can choose from a range of colors.
The following only applies to LibreOffice Writer: If you click the icon with a short-click while no text is selected, then the mouse pointer changes its appearance and is displayed as a paint can. Use this paint can symbol with the mouse key pressed to drag across a text area. This text area takes the selected color. The function remains active for as long as the icon is pressed, or until you click without dragging, or until you press the Escape key.
The following applies to all modules (LibreOffice Writer, Calc, Draw, Impress): Select the text that is to take another color, then click the color you want on the toolbar.