Purchase of domain and registration of brand name are two separate activities and only purchase does not give rights over the brand name.
For example: amazon.com domain is purchased by Amazon Technologies, Inc. However, someone may come up and purchase amazon.in, amazon.eu. amazon.info,etc.
However, once Amazon Technologies, Inc. registers brand “Amazon” with IPR department, then they will be rightful owner of the brand “Amazon” irrespective of domain name owner.

Yes, you may go for full domain name registration (say: amazon.com) or registration of the brand name (say Amazon)