Race is social, not scientific, and the parameters of the "races" have always shifted throughout recent history. The US Census places South Asians within the "Asian" category, alongside East Asians. Which intuitively makes no sense, but that's how it is according to the US Census. Censuses in other countries, like the United Kingdom, Brazil, Canada, or South Africa categorize people differently than the US, because all of these are subjective, not scientific. In Mexico, where the population has a mixture of Amerindian and European ancestry, people vary from very brown to very white, but the Mexican census considers "indigenous" only those that actively continue to live a pre-Columbian lifestyle and culture, which is a tiny proportion of the population.
South Asians (not to be confused with Southeast Asians) were indeed once considered "Caucasoid", according to an old "race classification system" that categorized everyone into only three races. That three-race classification is now regarded as anachronistic.
South Asians are actually a mixture of two "races"...the indigenous/original inhabitants, the dark-skinned Dravidians and the light-skinned Indo-Aryans who invaded the region a couple thousand years BC. This is why, today, southern Indians are darker than northern Indians; northerners have more Indo-Aryan ancestry, southerners have more Dravidian ancestry. It's also the reason for the linguistic divide in India. Northern Indian languages like Hindi, Bengali, and Gujarati are Indo-European languages, meaning they're related to European languages (English, Spanish, German, Russian, Greek, French, Italian, Dutch, Polish, Portuguese, Danish, Serbo-Croatian, etc), and completely unrelated to southern India's Dravidian languages (like Tamil, Kannada, and Malayalam).
Race is mostly a social construct. There's actually more genetic diversity within the Black "race" (people of Sub-Saharan ancestry), than among the rest of the human population. This is due to the fact that Sub-Saharan Africa has been populated by humans far longer than any other continent, thus more genetic variation among humans developed there. But, because we define race to a large extent by skin color (which is merely the product of human adaptation to climate), we group all of these people as a single "race".
In other words, race is mostly bullshit, entirely social based on very few selected genetic traits -those we're taught to see as most obvious- and not at all scientific.
All modern humans belong the same true race in a scientific sense (subspecies), Homo sapiens sapiens. Other races, like Homo sapiens idaltu, have been extinct for hundreds of thousands of years.
In American society, "race" is also a cultural identifier and your perceived race also determines how society treats you (i.e. discrimination), so that's why we ask people for "race" in the US Census or in surveys, job applications, etc.