Getting a bank account while a non-resident

Most banks require a residential address within Germany to open an account. Ask the bank of your choice for requirements and formalities.
The requirement for personal identity check can be fulfilled at a German embassy (for a fee).
An EC card can be used EU-wide (theoretically worldwide, but acceptance outside of the EU is rare).

That seems to be the case, it's fine to use my own bank account then? It is possible that I would be 1. Not allowed to open an account without an address and 2. Not use my bank account.

As Beppi said most banks require a residential address within Germany to open an account. Maybe if you had a business in Germany and was giving them a lot of your money from sales they might let you off by having an address in another EU country. But they like to know where you are, I think they want to make sure it's not being used for illegal activity (money laundering). You can only ask the banks >I guess?

Hi,
As others have tried to hint / workout: money-laundry law requires the bank to know its customer. This requires i.a. proof of identity and address. That can be done. The next question is, what the heck do you need a German bank account when you live in France? You have a French account. That is absolutely fine! You are inside the SEPA system which means "single euro payment area" or check out https://en.wikipedia.org/wiki/Single_Euro_Payments_Area