A Datamodule is for non-visual components only.
Besides a TPageControl you can also use a TNotebook which neither changes the form background color nor leaves a border. But the need to hide components is a good indicator that the form has become too complex. The correct way to handle this is by putting the components on TFrame components which are form-like classes and separate also the event handlers into their own unit. Ideally you insert the frame(s) into the mainform at runtime. The Lazarus IDE is a good example for working with frames. Another, less complex, example is the charteditor demo in folder components/tachart/demo/charteditor of your Lazarus installation.