<p>Recent advances in cough sound analysis using deep learning techniques enable smartphone-based respiratory disease screening suitable for self-management care in a home setting, yet their utility is limited by device heterogeneity, population diversity, and challenges in multimodal integration. We propose a device-invariant, multimodal deep learning framework that jointly models cough acoustics, demographic data, and symptom descriptions for multi-label classification of adult respiratory diseases. To address the issues of device effect, an adversarial branch is embedded in the audio encoder to enforce device-invariant feature learning, while an invariant risk minimization-augmented loss enhances robustness to non-structural shifts. To evaluate the effectiveness of our proposed method, a real-world, multi-center dataset containing over 10,000 cases spanning seven major respiratory conditions was curated. On the tasks of individual respiratory disease identification for chronic obstructive pulmonary disease (COPD), lower respiratory tract infection (LRTI) and pulmonary shadows (PS), our method achieves superior performance with the area under the receiver operating characteristic curve (AUROC) of 0.9698, 0.8483 and 0.8720, respectively. It also shows promising results in identifying the presence of comorbidities for 7 respiratory diseases with an overall AUROC of 0.8907. More importantly, extensive experimental results demonstrate our method mitigates the issues of device effect and facilitates the cross-device generalization for cough-based respiratory disease diagnoses. This work demonstrates a scalable and transferable AI-based approach for cough-driven respiratory screening, emphasizing the importance of multimodal fusion and robust representation learning in advancing clinical applicability.</p>