With advanced imaging, sequencing, and profiling technologies, multiple omics data become increasingly available and hold promises for many healthcare applications such as cancer diagnosis and treatment. Multimodal learning for integrative multi-omics analysis can help researchers and practitioners gain deep insights into human diseases and improve clinical decisions. However, several challenges are hindering the development in this area, including the availability of easily accessible open-source tools. This survey aims to provide an up-to-date overview of the data challenges, fusion approaches, datasets, and software tools from several new perspectives. We identify and investigate various omics data challenges that can help us understand the field better. We categorize fusion approaches comprehensively to cover existing methods in this area. We collect existing open-source tools to facilitate their broader utilization and development. We explore a broad range of omics data modalities and a list of accessible datasets. Finally, we summarize future directions that can potentially address existing gaps and answer the pressing need to advance multimodal learning for multi-omics data analysis.