There is only one way under the FCRA to compel them to provide documentation related to the alleged account, and that is to initiate an identity theft procedure.
Disputes require them to investigate, but verification only requires them to make a finding that the information is accurate. It does not compel them to provide supporting documentation.
By filing a police report (which the FCRA calls an identity theft report), you can send a copy to them and require that they provide a copy of "application and business transaction records in the control of the business entity." FCRA 609(e).
You can also send a copy of the police report to the CRA, and immediately get the information blocked from further inclusion in your credit report. FCRA 605B.
Proof of identity can simply be "personally identifying information that the business entity typicially requests from new applicants." FCRA 609(e)(2)(i).
A small price to pay to get both CR exclusion and copies of all business records pertaining to the alleged identity theft information.
Of course, you can always skip efforts based on your rights under the FCRA, and simply file a civil action.
Did you make an automobile purchase or a motorcycle purchase around that time possibly? When I bought my motorcycle I received three new credit cards a Suzuki one a honda one and a Yamaha one. I didn't ask for them but when they sent of for financing I got them all approved they stayed on my credit for like 5 years. I never used them but they were there? I'm just throwing that out there as another explanation possibly.