Yes, I believe that is possible for single entity in the Netherlands trading into the EU. I believe the business has to have a selling value threshold to register for VAT in an EU country, but I am sure they know that. I believe you can register as a one stop shop (OSS) or in each country individually and you would pay normal VAT for the Netherlands and then do a second VAT return for transactions in the destination country and claim VAT back on expenses spent in that country.
Localisations do not matter, it is a Netherlands legal entity registering for VAT in other countries. You dont need to share anything, it is one legal entity.
Ultimately speak to the customer to understand the detail, let the finance consultant configure it.