A bit of both, in essence.
Crossrail/Elizabeth Line will be operated, just as TfL Rail is now, by MTR Corporation (Crossrail) Limited. They are a National Rail operator and hence National Rail staff travel facilities will be valid on them, and travel on paper tickets will be subject to the National Rail Conditions of Travel.
If you have the 'Priv' discount set on an Oyster card, I would expect that, as with Thameslink, you will be charged the discounted fares when travelling on Crossrail so long as there is no Tube travel involved in your journey.
However, most of the 'core' stations are managed by London Underground. And there doesn't appear to be any indication that there will be paper National Rail only fares for travel the core stations (as there are between Farringdon and stations serve by Thameslink, for example). The core stations will also not be treated as London Termini, hence you won't be able to use a London Terminals ticket to travel through the core. You would need to buy a ticket to the other side of London.