Our objective is to investigate the inverse problem of identifying variable parameters in certain variational and quasi-variational inequalities. To this end we extend a trilinear form based optimization framework that has been used quite effectively for parameter identification in variational equations emerging from partial differential equations. An abstract nonsmooth regularization approach is developed that encompasses the total variation regularization and permits the identification of discontinuous parameters. We investigate the inverse problem in an optimization setting using the output-least squares formulation. We give existence and convergence results for the optimization problem. We also penalize the variational inequality and arrive at an optimization problem for which the constraint variational inequality is replaced by the penalized equation. For this case, the smoothness of the parameter-to-solution map is studied and convergence analysis and optimality conditions are given. We also discretize the identification problem for quasi-variational inequalities and give the convergence analysis for the discrete problems. Examples are given to justify the theoretical framework.