Yes, Belgium has a similar regulation regarding waste. However, the registration in Belgium is split up into 3 regions. For each region, a separate registration is required. For more information you can contact the 3 regions.

1. Flemish region
OVAM: Openbare Vlaamse Afvalstoffenmaatschappij
Stationsstraat 110,  B-2800 Mechelen, tel: +32 (0)15 284 284
Website: www.ovam.be 
E-mail:registratie.vervoerders@ovam.be 
 
2. Walloon region
SPW : Departement du sol et des Dechets Office Wallon des Dechets, Direction de la Politique des Déchets
Avenue Prince du Liège 15,  B-5100 Namur (Jambes), tel: +32 (0)81 336 531
Website: http://environnement.wallonie.be
E-mail: jeanyves.mercier@spw.wallonie.be

3. Brussels-Capital region
Leefmilieu Brussel
Gulledelle 100,  B-1200 Brussel, tel: +32 (0)2 775 75 11
E-mail: permit_agr@leefmilieu.irisnet.be