Re: Amar's a masculine Indian name, Sanskrit for "immortal", but it could easily be feminised into Amara if one chooses.
I agree, however, just to be clear, in India I haven't heard anyone choose to do so. In fact, most people would have to look up a dictionary for the feminine form of amar (the form without the feminine suffix was the form used in compounds, was always used as the common or collective gender, and, in any case, gender correspondence has weakened in modern Indian languages, except in substantive use), though they would probably indeed choose amara if forced to do so. In fact both amari and amara have been used in different periods in Sanskrit.
I do not know, but it is possible that amara in the technical senses of umbilical cord or afterbirth was probably more common than as the feminine form of amar in classical Sanskrit.
Given this, I think this name is unlikely to be of direct Sanskrit origin (a long history going back to Sanskrit is certainly possible). In any case, the first and second vowels in sanskrit are closed, not open (open means as in the English car) as in the last feminine suffix. I think in the name amara, the three syllables are the same except for length.
To me it sounds Persian (in which case it is related to the same root mR which gave the sanskrit name, and English words like mortuary, immortal, murder etc.), if it is not Semitic (I think I have heard it in middle eastern names) or Egyptian (Doesn't it appear in names of cities and temples?)