I believe the reason why you can on Pages and Files and can't on Custom Tables/User Registration is because there is a specific "Attachment" database table that links attachment files to Documents on the tree, where as Custom Tables and User Settings do not have that, nor the ability to set a Binary type field.
Forms have a Data Type "File" (binary), so that can accept a file.
So that being said, here are your options.
You can use a Form to capture all that information, and then either create the user manually from that (no coding required), or use the Global Event Hooks to capture the Form Submission, grab that submission's information and programatically set things up (store the file in the Media Library and put the path to it in a Text field for the picture).
Create a custom Form Control that uploads the given file they select to the Media Library, and returns the Url. This way on your Custom Data Table / User Settings, you can have a Text field "FileUrl" and put this custom control there.
Of the two, i would recommend #2.