Deep learning models are able tomemorize the training set. This makes them vulnerable to recovery attacks, raising privacy concerns to users, and many widespread algorithms such as empirical risk minimization (ERM) do not directly enforce safety guarantees. In this paper, we study the safety of ERM models when the training samples are interpolated (i.e., at interpolation) against a family of powerful black-box information retrieval attacks. Our analysis quantifies this safety via two separate terms: (i) the model stability with respect to individual training samples, and (ii) the feature alignment between attacker query and original data. While the first term is well established in learning theory and it is connected to the generalization error in classical work, the second one is, to the best of our knowledge, novel.Our key technical result characterizes precisely the feature alignment for the two prototypical settings of random features (RF) and neural tangent kernel (NTK) regression. This proves that privacy strengthens with an increase in generalization capability, unveiling the role of the model and of its activation function. Numerical experiments show an agreement with our theory not only for RF/NTK models, but also for deep neural networks trained on standard datasets (MNIST, CIFAR-10).