OP, most of the advice you have been given so far is either incorrect or bordering on nonsense.
The V5 is just a record of liability, whoever is named on the V5 is responsible for all fines, fees and taxes relating to the vehicle. The V5 is nothing to do with ownership. That's why when you lease or PCP a car you don't get the V5, it stays with the lease company.
You, as the person named on the V5, ARE responsible for the fine, and YOU WILL have to pay it.
There is a massive GDPR breach on the part of the garage for giving you the new owners details.
As long as you are the registered keeper YOU will be responsible for any speeding fines (NIP will be sent to you and you will be responsible if you don't fill it out) parking tickets, toll charges, the lot.
If the new owner has questionable morals, they might not be interested in fixing this issue, at the moment they are immune from all speeding fines and parking tickets. The private parking tickets are particularly troublesome as they take ages to sort out. This "small" issue has the potential to bankrupt you, trash your credit rating, have bailiffs or County Court officers at your door to sieze your property and basically ruin your life.
Simple solution is this. Fill out the new keeper section of the form, saying the vehicle is "in trade" at the dealership you used, send it off to the DVLA, they will send confirmation of change of keeper to you within 14 days. That sorts your first problem.
Next, pay the fine (you have no choice in this) and than pop into the dealership and ask to be reimbursed, if they refuse, your only option will be to take them to court, but you will win so don't worry about that.
Do not ignore this, do not discuss any further with the dealership.
You are.responsible for the fine, there is no way around this, the DVLA are correct and won't back down.