I assume the request for a registration number is for VAT purposes.
In many EU countries including the one I mainly work in an invoice needs to include the VAT registration number of the supplier as well as the client. Also, for business within the EU (above a certain amount) you need to fill in an EU sales list which includes all your EU suppliers and their registration numbers. So asking for a tax registration number seems like a reasonable request.
If you are not registered for VAT then there is no VAT registration number to give of course.
NINO is something completely different as far as I know, this relates to you personally and not your freelancing activities and should never be asked by a client.
UTR is used for your yearly self assessment and should not involve an overseas client.
(I am a freelancer in EU, non UK)