Automatic Language Identification (LI) or Dialect Identification (DI) of short texts of closely related languages or dialects, is one of the primary steps in many natural language processing pipelines. Language identification is considered a solved task in many cases; however, in the case of very closely related languages, or in an unsupervised scenario (where the languages are not known in advance), performance is still poor. In this paper, we propose the Unsupervised Deep Language and Dialect Identification (UDLDI) method, which can simultaneously learn sentence embeddings and cluster assignments from short texts. The UDLDI model understands the sentence constructions of languages by applying attention to character relations which helps to optimize the clustering of languages. We have performed our experiments on three short-text datasets for different language families, each consisting of closely related languages or dialects, with very minimal training sets. Our experimental evaluations on these datasets have shown significant improvement over state-of-the-art unsupervised methods and our model has outperformed state-of-the-art LI and DI systems in supervised settings.