Can my customer buy Azure through CSP from multiple partners apart from me ?

Yes, customers can choose to buy Azure through multiple partners at the same time. Each of those subscriptions are controlled and managed by the respective partners for the customer. The customer can choose to extend delegated permissions to manage resources for one or more partners on the subscriptions they own – typically in scenarios where the subscriptions comes from Open, Web direct or Enterprise agreement. In CSP the Partner has admin rights on customer tenant by default and the customer does not have admin privilege.

 For more information on buying CSP Azure subscriptions from multiple partners, refer here