Precise control in modern robotic applications is always an open issue due to unknown time-varying disturbances. Existing meta-learning-based approaches require a shared representation of environmental structures, which lack flexibility for realistic non-structural disturbances. Besides, representation error and the distribution shifts can lead to heavy degradation in prediction accuracy. This work presents a generalizable disturbance estimation framework that builds on meta-learning and feedback-calibrated online adaptation. By extracting features from a finite time window of past observations, a unified representation that effectively captures general non-structural disturbances can be learned without predefined structural assumptions. The online adaptation process is subsequently calibrated by a state-feedback mechanism to attenuate the learning residual originating from the representation and generalizability limitations. Theoretical analysis shows that simultaneous convergence of both the online learning error and the disturbance estimation error can be achieved. Through the unified meta-representation, our framework effectively estimates multiple rapidly changing disturbances, as demonstrated by quadrotor flight experiments. See the project page for video, supplementary material and code: https://nonstructural-metalearn.github.io.