The research on this Vigraha is proving to be simple and yet difficult.
A Goddess with four-arms and a serpent in each of the arms, who else can it be other than a Snake Goddess. Manasa is the first name that comes to our mind when you see a classical version of Snake Goddess.
Manasa worship is prevalent in Bihar and West Bengal. The iconography, form and style, rules out the possibility of this Vigraha being Manasa from Bihar or West Bengal. This site has an example of Manasa, a portable stone sculpture. It is included here for quick comparison.
A closer look at the Vigraha suggests it is possibly from Karnataka or nearby region. The indicators include the side tassels of her lower garment (You see those in Vijayanagar or post-Vijayanagar Vigrahas, especially that of Vishnu), the alloy and the absence of Bhadrapeetha.
One of the views expressed, in the book Buddhist and Brahmanical Sculptures in the Dacca Museum by Nalini K Bhattasali, is the name Manasa could be traced to the independent Goddess Manchamma or Mancha worshipped in parts of Karnataka and nearby areas of Telugu speaking region.
It is quite likely this is a classical portrayal of the Grama Devata (village deity), Manchamma, as worshipped in a specific location. A quick search shows several Manchamma temples in Karnataka.
Dr. Raghavendra Kulkarni is of the view this is a Grama Devata and he pointed out the presence of Manchamma Temple in Mantralayam. According to him, this is probably a Ishta devata (personal deity) and may not follow the established iconography. Dr. Kulkarni was kind enough to share some literature on Snake worship and the research continues.
The Vigraha is about 10 cm in height. The worship related wear and tear suggests date of nineteenth century or earlier.
The four-armed Goddess holding a serpent in each of her hands and a classical representation of a folk deity makes this special.
Note: When doing research on this Vigraha I came across references to Minoan Snake Goddess (from Greece) and its possible links to Manasa. Further reading showed Minoan figure was found without head and had only one arm (holding a broken curved stick). Somewhere/by someone she got transformed to a snake goddess holding two snakes in her two arms.
Now we have a cottage industry selling Minoan Snake Goddess figures and ‘enthusiasts’ are building theories about how Hindu Deities, Manasa, is linked to Minoan Snake Goddess. Ongoing effort to link Hindu deity to Roman/Greek Pantheon.