Thank you all for your posts!
We have informed the loan department but they want the police report, identity theft report, etc. The police officer sided with the dealership saying that we showed intent, which the lawyers said is an expected development in our neck of the woods. The lawyers also elaborated at length (at $250/hr) how unfair the local justice system is, frequently siding with large established businesses. Here are the details, though, I am glad you asked for them.
The car has a cosigner, our former babysitter, who we discussed demonstrating that we are her source of income at the time she was searching for a car (since she does not work for a hospital, Walmart, or similar, some letter, we thought, could be needed). Unfortunately, in an ESL mistake a word "cosign" was used. Yet we never went to a dealer, never signed anything or even saw a loan application, never had the car, did not know the amount or conditions of the loan until the bank has started sending us statements. We had purchased 2 cars from the same dealership, and they used our previous application, changed some numbers, had the babysitter also sign it, and submitted it to the bank like this! The babysitter told the dealership, we found out later from the police investigation, that the transaction was to be secret from the wife. Thus, the husband, a local doctor with impeccable credit, ended up a co-owner of her car and of her loan, the babysitter has fallen off the face of the Earth, is not making payments, and is tarnishing our credit. Luckily, we have the email from the dealer stating what they did (see original post). Originally, we assumed that they used us as her reference like that (replied "thank you" to email, thus - intent), but eventually it transpired that they committed us to a nearly 30K loan (thank goodness it was not a Porsche).
Now the loan department is likely to repossess the car and hold us accountable for the loan we never signed + all their expenses, seeing that the babysitter has nothing to take. Even if they don't repossess it, we are looking at years of destroyed credit. So a lawsuit of some sort is likely, it seems.
Also, the lawyer recommends having possession granted by the civil court and taking over the loan and car. Would be really great if we could avoid this.