If a company is registered in Europe and wants to conduct economic activity, then at registration it is assigned V.A.T. (analogue INN in the Russian Federation).
Value Added Tax is the registration number of the tax payer. It is valid only in the EU. and issued by the decision of the local tax authorities of any EU country, and only to companies that actually conduct commercial activities in that particular country. It is used in invoicing.
For example, for France, it will look like this: FRXXXXXXXXXXXX
Check VAT number on this site: http://ec.europa.eu/taxation_customs/vies/vatResponse.html

EORI (Economic Operators' Registration and Identification) is the base of enterprises that conduct foreign trade activities in the EU. Issued by the decision of the customs authorities in the operations of import, export and transit.
The number is needed for customs clearance of goods.
EORI consists of a 17-digit number. For example: FI1234567890XXXXX
You can check the EORI number on this site: http://ec.europa.eu/taxation_customs/dds2/eos/eori_validation.jsp?Lang=en

Those. if the company simply operates in the European market, it will have a VAT number. If the company exports, imports or performs transit operations with the goods, then it must also have an EORI numbers.