In short – Yes. For one, microchips are standardized around the world, and a vast majority of countries list it as a mandatory requirement to import your pet. Even if your destination does not list it as a requirement, the Animal Quarantine and Certification Service in India requires every pet being exported to be microchipped. If your pet isn’t, they will not grant you an export No Objection Certificate (NOC), without which your pet cannot leave the country with you.
Microchips for pets around the globe are standardized. The standard, 15-digit, non-encrypted ISO 11784/11785 are used and accepted worldwide. So yes, your pet needs a microchip to travel. Even if your travels are domestic, an extra safety net for the well-being of your pet is always welcome.