The letter actually says that "On Friday 7th June you were reported to TfL for an offence on a TfL public transport network." It doesn't say the offence took place then, unless that is splitting hairs. Maybe someone in a back office was compiling evidence against you and by 7th June they had completed their investigations and passed them on to TfL. Or it's a fishing trip. Maybe you should respond asking for more details since you do not remember any incident on that day / you did not travel on the TfL network that day, and that without further details you are not in a position to respond.
Standard TfL practice if they have collected digital evidence of fare evasion is to revenue block a commonly used station at a time they believe the perpetrator will be using it.
It is unheard of for them to contact someone without stopping them to gather details. If they had a registered Oyster card it’s possible through this, but what’s to say it’s up to date? Or than the registered person is the person using it.
To get a bank to give their details the BTP would probably need to get involved, as whether TfL is a ‘relevant authority’ is debatable, and banks in particular are very wary of privacy laws.
In summary it’s very unlikely this is a fishing expedition or a larger operation. I repeat my perilous advice to the OP, deny being stopped but don’t lie. If they have contacted you without stopping you they probably have evidence of a significant amount of fare evasion, and TfL would almost definitely take you to court regardless
