In religious studies, the faiths of Hinduism, Buddhism, Jainism and sometimes Sikhism are classified as Dharmic religions. This is not necessarily because they share origins but because they share themes and terminology. Usually they are regarded as distinct faiths in academia. They are only associated under the banner of having a some similarities. Islam, Judaism and Christianity are treated the same way under the banner of Abrahamic Religion. Daoism, Shinto and Confucianism are united similarly under the title Daoic Religion.
Siddhartha Gautama was born into a Hindu or secular family depending on who you talk to but the middle path that is Buddhism was not born out of the ideas of his family. Buddhism was a new revelation with a focus on inner cultivation and spiritual moderation. It retained some aspect of Hinduism but they are large and vague elements common to many unrelated religions (reincarnation, escape, etc)
Newer forms of Buddhism have also been influenced by East Asian Daoic faith. Chan Buddhism in particularly has been dramatically effected by Shinto and Daoism, taking it even farther away from Hindusim. Tell a typical of Japanese Chan Buddhism that they are Hindus and they will give you quite the funny look.
Jainism's absolute origins are somewhat mysterious in the same way as Hinduism's. However, its emphasis on asceticism and rejection of the Vedas separate it from Hinduism considerably. However, the two have interacted significantly and there may have been some crossover in beliefs.
Sikhism is a unique case. Many scholars view it as a combination of Hinduism, Islam and newer ideas. In this sense it is not an off-shot but a new syncretic faith. In this sense, it is both Dharmic and Abrahamic. However, most Sikhs consider their religion to be in a class of its own.
Parsis are members of the Indian Zoroastrianism community and the faith is not related to Hinduism or the other Dharmic religions. Zoroastrianism is a western religion, possibly the prototypical western religion. Still, Parsis are an integral part of India at this point.
Further complicating the issue is the fact that Hinduism actually describes a number of broad ideas and world views that could easily be described as separate faiths (Vaishnavas, Shaktas, Smartas). Some forms of Hinduism may resemble Buddhism and Jainism significantly in focus but often differ in methods and cosmology.
In another sense, Hinduness is like Jewish identity in that it can have an ethnic component. One can be a Muslim religiously but be ethnically Jewish (If my girlfriend and I had kids, this would be the case). Similarly, one could argue that people that are descended from ethnic Hindus will always be Hindu. I have heard this brought up often regarding Islamic Pakistan and arguments for it being returned to India. I don't know how multiethnic individuals like myself fit into this but I imagine it would be complicated due to the caste system.
As a poster above mentioned, the government of India states that most dharmic faith followers are Hindu but I don't think many will agree that a government is a spiritual or academic authority. For example, the U.S. lumps Middle Easterners, North Africans, Russians and Europeans as White even though many of those groups don't consider themselves as such. Also, Jainism was ruled not to be a Hindu faith in 2005 as per Supreme Court of India judgment, CASE NO.: Appeal (civil) 4730 of 1999 Bal Patil & Anr. vs. Union of India & Ors ;DATE OF JUDGMENT: 08/08/2005. This judgment may lead to a reconsideration of the other faith's legal status.
Then again, this is all about self-identity and that is an artificial mental element. So these religions are what the individual believer believes them to be. Personally, I believe the faiths to be distinct enough to qualify as separate, independent religions. However, I only study religion. I don't practice any of the dharmic faiths or any other faith for that matter.
From time as a religious studies major at U.C. Berkeley. Also from life experience as a secular/agnostic half-Pakistani.