Hi Abhi Jani,
No, it is not necessary to make an application of Trademark Registration for Company Name.
However, you can voluntarily apply for the Trademark registration. In practical aspects, it may happen that the products and services of the company are marketed by any other name. Hence the brand name is created in that name, which is being marketed by the company. Here, the product name is more important to the organisation. Hence, for that brand name, a company can apply for Trademark registration as an applicant.
The name of the company itself is unique and the same name is not allotted to any other body-corporate being any company or LLP as their name. Hence, to protect that name you are not required to apply for trademark. But, the same name can be used by any sole proprietor or Partnership firm and alike entities not being Body Corporate, to protect the usage of the same, you may apply for Registration under Trademark.