Yes, the operator number you receive from the RDW in the Netherlands is valid in all EU member states and in countries that follow European drone regulations (such as Norway, Switzerland, and Iceland). This is because the registration is part of European regulations (EASA), which are the same everywhere.
You register your drone in the Netherlands and receive operator number NLDxxxxxxx. When you then fly in France or Spain, you use the same number. However, you must comply with the local no-fly zones and additional rules in those countries.
Yes, your Dutch operator number is valid in all EU countries that follow EASA regulations. This means you don’t need to re-register abroad, but you must always comply with local flight regulations and restrictions.