Why should I give you my ID card

> First, no company should ask me for documents defining my identity

If by 'defining' you mean 'proving' then I'm afraid this is completely disconnected from reality, be it online or in the physical world.

Even leaving legal requirements aside, individuals and companies routinely enter into onerous contracts and it is perfectly reasonable to assert the identity of the other party beforehand, if only to be able to take legal action against them if needed.

The prime example for most people is obtaining a mortgage or any sort of loan. Would banks be willing to lend hundreds of thousands of pounds to "John Mclure" just because that's the name the applicant filled in? I don't think so.