Continuous, automated monitoring of laboratory mice enables more accurate data collection and improves animal welfare through real-time insights. Researchers can achieve a more dynamic and clinically relevant characterization of disease progression and therapeutic effects by integrating behavioral and physiological monitoring in the home cage. However, providing individual mouse metrics is difficult because of their housing density, similar appearances, high mobility, and frequent interactions. To address these challenges, we develop a real-time identification (ID) algorithm that accurately assigns ID predictions to mice wearing custom ear tags in digital home cages monitored by cameras. Our pipeline consists of three parts: (1) a custom multiple object tracker (MouseTracks) that combines appearance and motion cues from mice; (2) a transformer-based ID classifier (Mouseformer); and (3) a tracklet associator linear program to assign final ID predictions to tracklets (MouseMap). Our models assign an animal ID based on custom ear tags at 30 frames per second with 24/7 cage coverage. We show that our custom tracking and ID pipeline improves tracking efficiency and lowers ID switches across mouse strains and various environmental factors compared to current mouse tracking methods.