Recently, a direct method of the time--frequency approach, called the signal separation operator (SSO), which is based on sinusoidal signal approximation, was introduced to solving the inverse problem of multicomponent signal separation. In a very recent paper "Direct signal separation via extraction of local frequencies with adaptive time-varying parameters", the authors obtained a more accurate component recovery formula derived from the linear chirp (also called linear frequency modulation signal) approximation at any local time. However the theoretical analysis of the recovery formula derived from linear chirp local approximation has not been studied there. In this paper, we carry out the analysis of SSO based on the adaptive short-time Fourier transform (STFT). We study both the sinusoidal signal-based model and the linear chirp-based model, and obtain the error bounds for the instantaneous frequency estimation and component recovery. The error bounds are derived by studying the approximation to the STFT of each component and by the assumption of the decrease of the Fourier transform of the window function for STFT. These results provide a mathematical guarantee to the proposed adaptive STFT-based non-stationary multicomponent signal separation method. In addition, experiments are provided to illustrate the general theory.