We are often getting the same kind of questions in a first conversation. Just clarifying a couple of things in the list below.
EORI registration
One company has maximum 1 EORI in each customs territory (EU, UK,...). A non-EU company can obtain an EORI in a EU country without having a VAT registration in the EU. If they do have a VAT registration, the EORI registration can be done in a different country. An EU company will always have the EORI registration in the country of incorporation. Having an EORI doesn't create a need for any reporting. In Switzerland this is called UID.
VAT registration
One company can have multiple VAT registrations. But maximum 1 per country. Having a VAT registration creates the obligation for periodic reporting.