As far as I can tell, one can hold any beliefs or lack thereof, and there is no need to register your beliefs with the government per se. However, there are laws where religion is relevant, such as the Hindu personal laws such as the Hindu Marriage Act, 1955 or the Hindu Succession Act, such as stating who gets your stuff if you die intestate. These laws apply to Hindus, and since Jains, Sikhs and Buddhists are treated as legal Hindus (!), it applies to them; and
to any other person who is not a Muslim, Christian, Parsi or Jew by
religion unless it is proved that any such person would not have been
governed by the Hindu law or by any custom or usage as part of that
law in respect of any of the matters dealt with herein if this Act had
not been passed.
and don't ask me to interpret that "unless clause". That is, the Hindu laws do not apply to Muslims, Christians, Parsis or Jews. On the face of it, that would mean that Yazidis would be treated as Hindus, w.r.t. the subject matter of those laws. Registering a religion isn't relevant to the question: what matters is that the Indian Government decided to create these particular laws, and they have not created any Yazidi-specific laws.
There are currently no national laws prohibiting religious conversion, and I can't find the state laws. This article discusses such laws, noting that some states require a person to register their conversion. I seems that the restriction is on A converting B, and not on B converting sua sponte.