Monitoring high-dimensional data streams (HDDS) in industrial Internet of Things (IIoT) settings provides significant insights for online quality management. In multi-variety small-batch customized production, data distribution changes over time, rendering traditional batch offline learning methods ineffective, and real-time data streams processing often encounters challenges from different types of concept drift. To this end, this paper developed a novel online adaptive IIoT big data streaming analysis framework (OAIDA), aimed at providing timely reliable key quality characteristics (KQC) selection and quality prediction for customized manufacturing. Within this framework, we proposed an interpretable local causal feature selection method (LCFS), mining the underlying causal mechanisms behind data streams. Additionally, to address various types of concept drift, an adaptive ensemble model with real-time performance average weighted (AERPW) was designed, demonstrating superior robustness in evolving HDDS. Furthermore, a concept drift detector was integrated into OAIDA, establishing dynamic sliding windows upon detecting changes in data distribution, aiding in distinguishing time-domain and time-invariant characteristics for modular and customized parts among products, respectively. Experiments on synthetic and real-world data streams validate the effectiveness of the proposed framework in causal feature selection and online learning.