The channel was created for BT TV when Sky renamed its sports channels. Previously we had Sky Sports 1 & 2.
No Sky Sports Extra is a Sky channel as evidenced by the license they hold with Ofcom
So content would be down to Sky BUT the stated and advertised purpose according to BT of the channel is to enable coverage of the live Premier League matches that are not transmitted on Sky Sports Main Event for BT subscribers to the Sky Sports Main Event addon.
The Bt Customer orders the add to the BT TV through BT. So that would indicate issues with Sky Sports Extra should be directed to BT who should redress any shortfall of service with Sky and manage its (BT) customer dissatisfaction if the add on package falls short of the customers expectation.