Dialect and standard language identification are crucial tasks for many Arabic natural language processing applications. In this paper, we present our deep learning-based system, submitted to the second NADI shared task for country-level and province-level identification of Modern Standard Arabic (MSA) and Dialectal Arabic (DA). The system is based on an end-to-end deep Multi-Task Learning (MTL) model to tackle both country-level and province-level MSA/DA identification. The latter MTL model consists of a shared Bidirectional Encoder Representation Transformers (BERT) encoder, two task-specific attention layers, and two classifiers. Our key idea is to leverage both the task-discriminative and the inter-task shared features for country and province MSA/DA identification. The obtained results show that our MTL model outperforms single-task models on most subtasks.