Yes AD does this by default, similar to how Eagle does it (pre Fusion Electronics).
I've taken over a number of different client projects over the years that were created in Altium but was never provided the original libraries. I have always been able to export all the component symbols and footprints into my own libraries, and then fix all the designs to then refer to those.
I only have experience of Altium SE99, and Summer 09 thru AD19, so I don't know if anything has changed in AD20/AD21.