To want to leave my Internet provider early.

[ImNotJoeMyNameIsHarry]

I'm in contract with them for 12 months, we're about 3 months in. They also do mobile contracts. Me and DP live in a house that was converted into flats so on some systems we are down as "number 19" rather than "flat c, number 19".
I've checked my bank and I have been charged for a mobile contract. When I've called them up they've said it's someone in my household has set up a contract and because I'm already in contract they added it to my direct debit so they don't have to put in bank details for the same address.

They don't even get why I'm annoyed or why it's a security issue. And this could be anyone, according to the election role theres about 15 people in our household.

But they won't refund, just say to get the person to call up and change the bill over. I don't want to be in this Internet contract if I can get billed for someone else. AIBU to be majorly pissed off at them or should I just cancel the bill and hope for the best.

This is absolutely shocking. I'd be threatening them with which, watchdog, even the police/action fraud.

Demand to speak to a manager. Ask why on earth they would do this without your consent even if you were within the same household. That's surely a huge red flag anyway. I can't believe this would be the case for student households, etc

Tell them that you consider this a fraud risk and themselves enablers.
Speak to your bank and see if they can claw the money back, demand your address be recognised as an independent entity and explain that if they cannot do that you will be looking for compensation and to end your contract.

I did try to ask they just pass me onto a random person. I also spoke to someone who said it might be my contract as the sales team set up contracts without the customer agreeing to! As soon as I work out how to edit my recorded calls I might just shame them. Also I found out my neighbour tried to set up a contract for his Internet days after we got ours and they charged him but refused to do it as therequested was already an account at the address. So he had to fight to get that back.