This is a false assertion. I have gone through sample birth certificates of multiple states in India, I did not find any of them mentioning caste. Caste certificate is a separate document, issued only upon application, which can then be used as proof for applying for reservation benefits.

Ref https://www.nriway.com/nri-service/nri-birth-certificate https://www.indiafilings.com/learn/caste-certificate-in-indi...