Who owns customer research? is a misleading question.
A better one is: Who can aggregate customer research and plan actions?
Research does not end with someone sitting down with a customer and asking a bunch of questions. Research is about putting the pieces together, identifying patterns, anticipating trends, sharing the knowledge and the insights, and eventually enabling everyone interested to access all this at their own convenience.
If you do not have someone responsible to aggregate customer research, you are not doing customer research at all.