India has ancient Jewish and Christian communities, and, in addition, has a lot of recent conversions. It also has an Anglo-Indian community arising from the colonial period. The names in these communities are often European and Biblical, and I suspect this to be the case fpr Annie and Achsah.
Madhu, of course, is an old Indian name, an Indo-european in origin (an English cognate is mead). It means honey or sweet, and was the generic symbol for all that was good, pleasant, and proper. I haven't heard it as a male name too often, probably because it was the name of a pretty unpleasant character in mythology. As a name, it often is part of a bigger word like madhusUdan, `the slayer of madhu', an epithet of kRSNa whom I describe explaining Brijesh last month. The sUdana bit comes from a word meaning to put in order or arrange according to taste, often an euphemism for removing or killing.