No but have a right to know there name if there is a dispute, anyone not displaying a name badge I think is breaking rules. As for posting an ID is the original poster mad.
No, they don't. We are under no obligation to give out our name at all. And no driver at my TOC (or many others) wears a name badge.
If a passenger demands my name I will offer them my depot and diagram number. They can also trace me by the train I'm driving (I will provide the headcode if they want) but my name is not required for any reason.
In this day of social media I will never give out my name. I don't want to see my name plastered all over Twitter or Facebook with some false allegation or derogatory comments mad about me followed by equally nasty comments from other people.
As for the Op, a train drivers licence looks pretty similar to a car licence. A credit card style piece of plastic with some sort of company logo design on it. It will contain your photo, name, depot, payroll number and expiry date. Many will contain on the front or back traction signed, PTS competency (ie non electrified, AC/DC etc). Any driver qualified to instruct will have this on their cars along with normally a green stripe at the top or bottom to act as a cab pass for the purpose of instructing and assessing.
The bulk of the licence is on paper kept on file and contains your route card, medical validity, corrective vision authority if applicable along with safety of the line record (sort of endorsements to crudely compare to the DVLA).