See holding an AADHAR card does not makes you an Indian citizen or give you Indian citizenship. Citizenship is granted explicitly by the Government of India and they provide you with some documents stating your Indian nationality. So, it all depends on which passport you hold or which country acknowledges you as it's citizen.

After doing these procedures and conforming with the Government of India, you may proceed with your NEET registration.

Hope I have answered your question. All the best!