The other problem is that without a reliable primary key, it's much easier for correct information about different people to be incorrectly mixed. While this is not as bad as, say, when it hits a sex offender database or terrorist watch list but it does happen and would go down dramatically if everyone could just use something like a SSN as the link instead of trying various fuzzy heuristics and hoping for the best.
If you live in the US you should consider filing every block available to you so for the banks you don't use, you can start the discussions with asking why they were illegally pulling your credit record or giving credit without it.
In Europe, it looks like banks are petitioning to be allowed to verify identity over camera phones using an independent third party. That is still better than the US, since you can probably force them to retrieve that record of "you and your id" fairly early in a dispute.
None. The credit reporting agency should be verifying any facts they intent to include in a report. If the are including unverified hearsay, the their report may be libel[1].
Also, your own ID card wouldn't fix someone else with either no ID or fraudulent ID doing something in your name. If the bank gave a loan to someone else, that's their problem, not yours.
[1] I'm aware that regulatory capture has granted the agencies immunity from slander/libel laws. Just because unethical behavior has been legalized doesn't make it the victim's responsibility to work around libelous claims.