Feature (logger): add tests for child logger attributes

[saragerion]

Description of the feature request

• [ ] Validate that a parent logger's persistent attributes are passed on to the child logger's persistent attributes via new/updated tests
• [ ] Improve documentation to make current behaviour clear

Problem statement

As brought up by @ijemmy during the internal beta testing, the current behaviour of the child logger and related attributes is not 100% clear. There is an opportunity to make the functionality clearer in the docs.

Summary of the feature

N/A

Code examples

Example of piece of code that can be updated: https://github.com/awslabs/aws-lambda-powertools-typescript/blob/main/packages/logger/tests/unit/Logger.test.ts#L750

Benefits for you and the wider AWS community

Describe alternatives you've considered

Additional context

N/A

Related issues, RFCs

N/A