It's certainly not GDPR's fault
The law is pretty clear, although it will not specify what prove of identity you need to provide it does leave the option open to request a copy of your ID (preferably not via e-mail for obvious security reasons).
The fault here squarely lies with the companies that have implemented the requirements partially, maybe are afraid of exceeding time limits or have obtained bad advice.
Should the law mandate how it must be done? No, as there are other laws to mandate what a prove-of-identity means and there is enough guidance available.
Besides if you know all that information from your girlfriend, there is probably no rat's chance in hell you could not have pulled this off.
Nice in showing at least the data providing part of the law works, okay it wasn't the right person. Btw, does she actually know you have her login credentials of certain website?
As for the information this person obtained: that's precisely the goal of the law (article 15 to be precise), to obtain a copy of all data a company holds of you.
An And yes that may include very sensitive information.
It shows nicely that one company has stuff they shouldn't have had in the first place, if you can believe this information about that given in this talk and she indeed hasn't heard of the company.
Btw, he precursor of the GDPR, at least in the Netherlands, already had the right to obtain a copy of information a company held of you anyway. So in that respect it may be less new than suggested.