Vanishing moments and approximation orders of basis functions are both the significant important properties in wavelets. The traditional customized multiwavelet construction methods could only change either vanishing moments or approximation orders. Meanwhile, the time-frequency characteristics and waveform difference among the new multiple basis functions are quite small, leading to the difficulty for efficiently adaptive extraction and identification of complex dynamic faults. Thus, the customized multiwavelets by the hybrid method combining two-scale similarity transform with lifting transform are proposed. Using the linear and nonlinear combination of multiple scaling wavelet functions to extend the construction space, the method could both obtain the multiple scaling functions with the high approximation orders and the multiple wavelet functions covering the multiple vanishing moments, which enhance the regularity, smoothness and the capabilities for signal approximation and local position, and improve the signal analysis precision. It provides the adaptive basis functions with super properties and a promising tool for the weak and compound fault feature extraction and identification. For the customized selection of adaptive basis functions, the improved local spectral entropy minimization rules are proposed, classified into the typical shaft, gear and rolling bearing faults, which simplify the fault modes. The engineering applications showed that the method could effectively identify the bearing weak inner-race damage disturbing by the complex background noise, and successfully diagnosis the impact and rubbing compound faults with multiple features from the thrust splint of gearbox.