With ongoing automation and digitization of the electric power system, several Phasor Measurement Units (PMUs) have been deployed for monitoring and control. PMU data can have multiple anomalies, and many of the researchers in the past have concentrated on training machine/deep learning algorithms offline for anomaly detection over PMU data (i.e., not in real-time). These machine/deep learning algorithms, when trained offline on a sample rather than a population of the dataset, fail to consider the dynamic behavior of the power grid in real-time, resulting in low accuracy. Considering the dynamic behavior of the power grid (e.g., change in load, generation, distributed energy resources (DERs) switching, network, controls), the definition of data anomalies varies in time and requires online training. A fundamental challenge is to enable online (i.e., real-time) training of machine/deep learning algorithms for anomaly detection over streaming PMU data. While machine/deep learning is often desirable to manage data streams, training a deep learning algorithm over streaming PMU data is nontrivial due to changes in data statistics caused by dynamic streaming data. This article proposes PMUNET: a novel device-level deep learning-based data-driven approach for anomaly detection, localization, and classification over streaming PMU data, using online learning and multivariate data-drift detection algorithm. Two variants of PMUNET, Dynamic data Change Driven Learning (DCDL) and Continuity Driven Learning (CDL), are proposed and compared. DCDL aims to train the deep learning algorithm whenever the definition of anomaly changes due to the power grid dynamics. On the other hand, CDL continuously trains the deep learning algorithm over the PMU data-stream. The experimental results verify that DCDL outperforms CDL and other efficient anomaly detection methods over multiple events such as faults and load/ generator/capacitor/DERs variations/switching for IEEE 14 and 39 Bus test system as well as real PMU industrial data. The result verifies that DCDL variant of PMUNET improves over existing approach with a gain of 2% - 10% in terms of accuracy, false-positive rate, and false-negative rate.