The train companies operate trains, not lines. For example; CrossCountry’s service runs on track shared with other passenger companies and freight from Penzance up to Scotland.
Great Western Railway operates the Great Western franchise which incorporates much of south wales and south west of england - Including the Great western mainline. Abellio Greater Anglia operates most services on the Great Eastern Mainline
As for permissions I don't know if the crossrail core section has exclusivity granted to TFL (or its subcontractor MTR Corporation), given it's run as a concession this may be the case.