Neural oscillators that originate from second-order ordinary differential equations (ODEs) have shown competitive performance in learning mappings between dynamic loads and responses of complex nonlinear structural systems. Despite this empirical success, theoretically quantifying the generalization capacities of their neural network architectures remains undeveloped. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper probably approximately correct (PAC) generalization bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating the uniformly asymptotically incrementally stable second-order dynamical systems are derived by leveraging the Rademacher complexity framework. These bounds are further extended to the squared Wasserstein-1 distances between the probability measures of quantities of interest calculated from target causal operators and the corresponding learned neural oscillators. The theoretical results show that the estimation errors grow polynomially with respect to both MLP sizes and the time length, thereby avoiding the curse of parametric complexity. Furthermore, the derived error bounds demonstrate that constraining the Lipschitz constants of the MLPs via loss function regularization can improve the generalization ability of the neural oscillator. Numerical studies considering a Bouc-Wen nonlinear system under stochastic seismic excitation validates the theoretically predicted power laws of the estimation errors with respect to the sample size and time length, and confirms the effectiveness of constraining MLPs' matrix and vector norms in enhancing the performance of the neural oscillator under limited training data.