An important challenge in metric learning is scalability to both size and dimension of input data. Online metric learning algorithms are proposed to address this challenge. Existing methods are commonly based on (Passive Aggressive) PA approach. Hence, they can rapidly process large volumes of data with an adaptive learning rate. However, these algorithms are based on the Hinge loss and so are not robust against outliers and label noise. Also, existing online methods usually assume training triplets or pairwise constraints are exist in advance. However, many datasets in real-world applications are in the form of input data and their associated labels. We address these challenges by formulating the online Distance-Similarity learning problem with the robust Rescaled hinge loss function. The proposed model is rather general and can be applied to any PA-based online Distance-Similarity algorithm. Also, we develop an efficient robust one-pass triplet construction algorithm. Finally, to provide scalability in high dimensional DML environments, the low-rank version of the proposed methods is presented that not only reduces the computational cost significantly but also keeps the predictive performance of the learned metrics. Also, it provides a straightforward extension of our methods for deep Distance-Similarity learning. We conduct several experiments on datasets from various applications. The results confirm that the proposed methods significantly outperform state-of-the-art online DML methods in the presence of label noise and outliers by a large margin.