[jsty]

IANAL, just currently wading through GDPR material.

As I see it the most relevant processing conditions for companies offering a service and storing / processing data without gaining explicit consent are likely to be 6(1)(b) - Processing is necessary for the performance of a contract with the data subject or to take steps to enter into a contract 6(1)(c) - Processing is necessary for compliance with a legal obligation

My understanding is that these are far from a blank cheque to store / manipulate arbitrary personal information. Specifically, the storage and use of data in question must be provably fundamental to either provision of the relevant service in (b), or meeting legal obligations in (c).

So yes, a company providing you a service will gain the right to store certain customer details demonstrably necessary to provide that service - say hosting your email. It won't however allow arbitrary use of such data to e.g. provide targeted advertising, since such use is not fundamentally required for performance of the service. This would require a specific opt-in (and from what I recall, a failure to opt-in cannot interfere with the provision of said service - not so clear on this however).