Causality analysis is a powerful tool for determining cause-and-effect relationships between variables in a system by quantifying the influence of one variable on another. Despite significant advancements in the field, many existing studies are constrained by their focus on unidirectional causality or Gaussian external forcing, limiting their applicability to complex real-world problems. This study proposes a novel data-driven approach to causality analysis for complex stochastic differential systems, integrating the concepts of Liang-Kleeman information flow and linear inverse modeling. Our method models environmental noise as either memoryless Gaussian white noise or memory-retaining Ornstein-Uhlenbeck colored noise, and allows for self and mutual causality, providing a more realistic representation and interpretation of the underlying system. Moreover, this LIM-based approach can identify the individual contribution of dynamics and correlation to causality. We apply this approach to re-examine the causal relationships between the El Niño-Southern Oscillation (ENSO) and the Indian Ocean Dipole (IOD), two major climate phenomena that significantly influence global climate patterns. In general, regardless of the type of noise used, the causality between ENSO and IOD is mutual but asymmetric, with the causality map reflecting an ENSO-like pattern consistent with previous studies. Notably, in the case of colored noise, the noise memory map reveals a hotspot in the Niño 3 region, which is further related to the information flow. This suggests that our approach offers a more comprehensive framework and provides deeper insights into the causal inference of global climate systems.