Mobile application classification is essential for advanced network management and application-based QoS policy enforcement in future, AI-enhanced, beyond 5G and 6G mobile networks. This article proposes to use AI methods to categorize applications as functional types (e.g., Video, Audio, Browsing) despite encryption and limited labeled data. We tackle these challenges through unsupervised representation learning, which maximizes the use of abundant unlabeled data in mobile networks. Due to the distributed nature of beyond 5G and 6G networks, we use this method in federated learning scenarios and compare it to the centralized ones. Our findings highlight that unsupervised learning improves model performance, especially with scarce labeled data. Additionally, federated learning provides effective results as compared to centralized methods.