Cough acoustics contain multitudes of vital information about pathomorphological alterations in the respiratory system. Reliable and accurate detection of cough events by investigating the underlying cough latent features and disease diagnosis can play an indispensable role in revitalizing the healthcare practices. The recent application of Artificial Intelligence (AI) and advances of ubiquitous computing for respiratory disease prediction has created an auspicious trend and myriad of future possibilities in the medical domain. In particular, there is an expeditiously emerging trend of Machine learning (ML) and Deep Learning (DL)-based diagnostic algorithms exploiting cough signatures. The enormous body of literature on cough-based AI algorithms demonstrate that these models can play a significant role for detecting the onset of a specific respiratory disease. However, it is pertinent to collect the information from all relevant studies in an exhaustive manner for the medical experts and AI scientists to analyze the decisive role of AI/ML. This survey offers a comprehensive overview of the cough data-driven ML/DL detection and preliminary diagnosis frameworks, along with a detailed list of significant features. We investigate the mechanism that causes cough and the latent cough features of the respiratory modalities. We also analyze the customized cough monitoring application, and their AI- powered recognition algorithms. Challenges and prospective future research directions to develop practical, robust, and ubiquitous solutions are also discussed in detail.