"I've also been told that your second language in school (which is taught in English, a neutral language) is determined by your race: Mandarin for Chinese, Malay for Malaysians, and Tamil (!) for Indians."

Remember though that Singapore is bilingual, so the "second language" will be the primary language spoken at home (although some Chinese may speak Hokkien or another dialect instead, and many families speak mainly English).

Roughly around 60% of Indians in Singapore are Tamil (mostly from Tamil Nadu, Malaysia, and Sri Lanka), so that could be one reason the government chose Tamil as the "second language" of Singaporean Indians.