StarryK,
Your short explanation of prepaid cards for cell phones is the same as for the debit cards I mentioned.
The difference is that these are real bank cards.
And they can be boought at real banks, not credit card companies or local stores.
Please check out the picture.
It includes the card, agreement, and a paper that original had the card glued to it.
Notice how it says "Key Possibilities Card", shows me as a Keybank Customer, and shows Keybank's street address.
There is nothing specifically of mine of the card. It is not attached to an account and is bought at a local bank.
There is no need to have an account to get this type of card. However, if you buy it at a bank, you may need to provide proof, such as a social security number (that is what this bank requires).
I have no idea what proofs of identification people in Europe use.