Would that mean opening up another account with another bank?
No
It is possible to get a credit card without it being associated with a bank.
For example years ago hotel chain I had a points card with offered a credit card, I applied, the card is now Barclaycard but I do not have a bank account with Barclay Card.
More recently I applied for a Virgin credit card, I have never actually activated it because it requires app activation (I have enough apps on my phone), I do not have an account with Virgin.
The Barclay card is paid from a non-Barclay bank.
Capital One offer credit cards, I don't think Capital one offer any sort of bank account - the payment would just go from your normal bank account. I also think some high street banks will let you have a credit card without banking with them (but I don't have any experience of this apart from Virgin).
I have paid adverts from my 'non-bank-account' credit cards for years, all that shows on my actual bank account each month is "Payment to whatever credit card"
A very long time ago I had a Sainsbury's credit card, just looked them up they seem to still offer them they might do them without you having to open a bank account with them (I didn't but it was 20+ years ago). You just set up the monthly payment from another bank.