This paper delves into the intricacies of synthetic data, emphasizing its growing significance in the realm of finance and more notably, sustainable finance. Synthetic data, artificially generated to simulate real-world data, is being recognized for its potential to address risk management, regulatory compliance, and the innovation of financial products. Especially in sustainable finance, synthetic data offers insights into modeling environmental uncertainties, assessing volatile social and governance scenarios, enhancing data availability, and protecting data confidentiality. This critical review attempts first ever classification of synthetic data production methods, when applied to sustainable finance data gaps, elucidates the methodologies behind its creation, and examines its assurance and controls. Further, it identifies the unique data needs of green finance going forward and breaks down potential risks tied to synthetic data utilization, including challenges from generative AI, input quality, and critical ethical considerations like bias and discrimination.