Yes all the institutes you mentioned offer animation and game design course. IIT Bombay offers Animation design as full time 2 years course.

Eligibility criteria is Bachelors Degree in Design/Engineering/Architecture/Interior design: 10+2+4 and also offers a four year professional program BFA with entry after 10+2.

VEL TECH offers a professional four year Gaming and Animation undergraduate course.

Vellore Institute of Technology (VIT), Bhopal offers a professional four year Gaming technology under B.Tech computer science engineering.

Many colleges in bangalore such as Jain university, Alliance university also offer Gaming and Animation courses.

Refer the below links for further information.

https://vit.ac.in/files/VITEEE-2024-information-brochure.pdf

https://www.careers360.com/colleges/industrial-design-centre-indian-institute-of-technology-mumbai/mdes-animation-design-course

https://www.veltech.edu.in/admission/