It is Sanskrit, and is the name of an asterism: a particular 13 1/3 degrees along the ecliptic named after a prominent star or constellation within it, in this case the constellation Hydra. Currently (i.e. for the next 1000 years or so), a full moon around the beginning of January is around this asterism.
The words representing these asterisms are very old: AshleSA' appears already in the Vedic literature and so is definitely more than 2500 years old, but probably not as a human name (in fact the male form had a slightly negative connotation: sometimes used for spirits that can grab you). The etymology involves the particle A' which qualifies verbs with a meaning of nearness, towardness, or completeness followed by the verb shliS, to join (metaphorically also to result in) or to connect, to embrace etc. AsleSA is the feminine of a noun meaning intimate contact, embracing, clinging or entwining.
The forms ashleSA and AshreSA are also attested in Sanskrit, probably as variants of the above.
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