You have similarity measures like mutual information score and generally comparing correlations and distributions. You can also A/B test for specific use cases

You have similarity measures like mutual information score and generally comparing correlations and distributions. You can also A/B test for specific use cases, for example train a model on the real and the synthetic data and compare relevant metrics. You can see some of these illustrated on eg: https://hazy.com/blog/2020/03/23/synthetic-scooter-journeys https://hazy.com/blog/2020/03/23/synthetic-scooter-journeys