Well perhaps we are back to WP filter and action hooks. Although as some users DO use DAP apparently successfully I am not sure what to say. But – so you are armed with information I can at least describe the process. In case you do not know it…
WordPress institutes a system of what are called ‘hooks’ which is the method used by plugins to integrate with the WP core. In a way you can think of them as events. When a user registers (using the WP system) it fires an ‘action’ hook that broadcasts to any plugins listening that there is a new user. SP is listening and when that event happens, creates the extra data necessary for that user to be a member on the SP forums.
If – of course, an alternative method of registering a user is employed such as a membership plugin then it is imperative that it also fires the same event hook that Wp would have fired.
On the SP side, new registrants will, by default, be placed in the SP ‘members’ user group although you can set the defaults in the forum admin > Options > Member settings panel.
If all this is not happening, then the first thing to look at is if the registration plugin is actually firing this event hook.