I live in the EU and work in data; there is no requirement to register as a data provider if you're working with customer data. Further, there will not be such as requirement under the GDPR either to my knowledge.
If you're a "data controller," then you may be required to register with the local DPA (Data Protection Authority). In the UK, for example [1]:
"The Data Protection Act 1998 requires every data controller (eg organisation, sole trader) who is processing personal information to register with the ICO, unless they are exempt. More than 400,000 organisations are currently registered."
Whether this registration achieves anything useful is another question.
