Every government veterinary college offers admission either through the state quota (85%) or the All India quota (15%).

For the All India quota, you have to appear for NEET while for the state quota, there are different states conduct their separate entrance exam for veterinary.

If we talk about Tamil Nadu, you can get admission based on your 12th marks if applying through the state quota. If you are applying through the All-India quota then you must have a valid NEET score. 

Now coming to your question you can get the domicile of Tamil Nadu if you are born here. It doesn't matter where you have studied. If your parents are Tamilians, you can get a certificate and you can apply through the state quota.