Are the opportunities the same for a vet passing from a private veterinary college and that from a government college (like TANUVAS, Bombay, and GBPUAT)?

Yes, because regardless of whether you graduated from a private or government-approved college, you will receive a registration number to practice from the state registration authority in the state where the college is located.

However, because of its strong alumni base, the government veterinary institutions have a better industry reputation.

However, only a few private colleges in the north are performing particularly well. 

Also in the current industry, most recruiters are concerned about specialization. They will be more concerned regarding your master’s degree. So, focus on your studies and try to join a reputed university for masters.