There is no doubt in some remote corner of India some party worker might ask people to swear on their kids, gods and what not. But let me confirm that it is not even a phenomenon. Mostly figure of speech. OP used graphic details as if it is some cult practice "For Hindu voters, a plate with oil lamp, an idol, a mixture of auspicious powder is used."

Either the party worker has to be naive to believe that such ritual has any value or the people have to be completely stupid. Poor people are desperate and will happily take whatever money that comes their way and will not care too hoots about swearing on Gods. Remember if you can do a ritual to swear on God another one can invent a ritual to free you from that oath.