I have paid about fourty dollars for aurora database serverless, without using it. Though to pay for aws services, i wonder how this is possible, when i do not use the database myself at all?
Does serverless not mean, that i just pay for usage?
Why do I pay for my serverless aurora database v2 when not using it?
Because you are using it - even if you aren't actively using it to service requests and queries, you're still paying for it to be operationally available. Think of it as being like an AWS VM: you can pay for it by-the-minute (or second?) but you have to pay for it so-long as it's running, and not only for the times when you have an active shell connection in your terminal.
AWS's documentation says that there is a minimum purchase of "0.5 ACUs" of Aurora V2 capacity, which is charged to your account for every second it is running:
Each ACU is a combination of approximately 2 gibibytes (GiB) of memory, corresponding CPU, and networking.
[...]
At any moment in time, each Aurora Serverless v2 DB writer or reader has a capacity. The capacity is represented as a floating-point number representing ACUs. The capacity increases or decreases whenever the writer or reader scales. This value is measured every second.
[...] The smallest Aurora Serverless v2 capacity that you can define is 0.5 ACUs.