Programs implemented in various programming languages form the foundation of software applications. To alleviate the burden of program migration and facilitate the development of software systems, automated program translation across languages has garnered significant attention. Previous approaches primarily focus on pairwise translation paradigms, learning translation between pairs of languages using bilingual parallel data. However, parallel data is difficult to collect for some language pairs, and the distribution of program semantics across languages can shift, posing challenges for pairwise program translation. In this paper, we argue that jointly learning a unified model to translate code across multiple programming languages is superior to separately learning from bilingual parallel data. We propose Variational Interaction for Multilingual Program Translation~(VIM-PT), a disentanglement-based generative approach that jointly trains a unified model for multilingual program translation across multiple languages. VIM-PT disentangles code into language-shared and language-specific features, using variational inference and interaction information with a novel lower bound, then achieves program translation through conditional generation. VIM-PT demonstrates four advantages: 1) captures language-shared information more accurately from various implementations and improves the quality of multilingual program translation, 2) mines and leverages the capability of non-parallel data, 3) addresses the distribution shift of program semantics across languages, 4) and serves as a unified model, reducing deployment complexity.