Biomedical studies often collect multivariate event time data from multiple clusters (either subjects or groups) within each of which event times for individuals are correlated and the correlation may vary in different classes. In such survival analyses, heterogeneity among clusters for shared and specific classes can be accommodated by incorporating parametric frailty terms into the model. In this article, we propose a Bayesian approach to relax the parametric distribution assumption for shared and specific-class frailties by using a Dirichlet process prior while also allowing for the uncertainty of heterogeneity for different classes. Multiple cluster-specific frailty selections rely on variable selection-type mixture priors by applying mixtures of point masses at zero and inverse gamma distributions to the variance of log frailties. This selection allows frailties with zero variance to effectively drop out of the model. A reparameterization of log-frailty terms is performed to reduce the potential bias of fixed effects due to variation of the random distribution and dependence among the parameters resulting in easy interpretation and faster Markov chain Monte Carlo convergence. Simulated data examples and an application to a lung cancer clinical trial are used for illustration.