I have picked up the idea that there is a plan for a 'new generation' Oyster Card (TfL issued) which will function the same way (as far as the Oyster/Contactless system is concerned) as a Bank/Financial Institution Contactless card - but at the end of day, after the day's 'bill' has been calculated, it will be deducted from the holder's Oyster balance stored in the backend system (rather than on the card). Or there is the possibility that it could be linked to a bank, etc. account from which funds could be drawn. (This is I think what the GW Bristal area PAYG system is doing).
The advantage - for the railway/TfL - is that it is THEIR card, so they can set its conditions of use, such as only to be used by the issuee (registered holder), which they cannot for a bank card, as the bank's conditions apply.