>> except in some special circumstances (e.g. childrens' toys)

> Don't nearly all electronics fall under it too, this is the second item after children toys in the link you posted.

No. That page just says: There are tons of EU laws about everything, including electronics. By putting a CE mark in your product, you certify that your product complies with all applicable EU laws.

In most cases, this certification can be a self-assessment, meaning that you can just assign the CE mark by whatever internal process you deem sufficient. I.e. you are allowed to rubber-stamp, but of course you are fully responsible for all consequences.

In some cases, for example childrens' toys, but only for some electronic components (electronic toys, medical stuff, high-voltage components, components that emit EM radiation above a certain threshold, ...), self-assessment isn't enough. In those cases, you need to obtain a certification from a notified body, which is an external organisation doing some design review and testing, as required by the respective laws.

It's all laid out further below under "How to obtain CE marking" and "Do you need an independent assessment?"