Obesity is a multifactorial disease. There are many contributing factors inducing epigenetics, environment, diet, psychology, social groups, hormones, activity levels etc - and maybe the microbome. However a lot more research and understanding needs to be done to suggest a causation between a less diverse microbome and obesity. Our understanding of what constitutes a healthy versus unhealthy microbiome is still evolving

Lastly, the science on probiotic supplementation for healthy individuals is weak at best. And quality control is lacking. I'd save your money and just eat some fermented foods like pickles, yogurt, kimchi etc.