In much of the world -- and in game programming generally -- software development careers are mostly a meritocracy. Your progress and advancement is based on the merits of what you have done already. Accomplish more, do it well, and more responsibility (and potentially power as well, assuming you are managing your career mindfully) will be added to you.
Getting your first job in the industry is frequently based on something else. Often it is the 'catch 22': the difficulty of needing work experience so you can get the job, but in order to get work experience you need to have had the job in the first place. Because you have no completed projects, nothing to judge your merits in the meritocracy, having a degree is frequently used as a guide to what you know. Completing a degree program shows you have at least a basic set of skills and knowledge, and can stick with a long project.
After you've been in the industry a while your work history makes it easy to get the jobs, but your academic history plays a part of salary negotiation. Generally those with more education can demand more pay than the similarly-experienced person with less education.
The level of education required is location dependent. Some areas of the globe have strict requirements, others have practically no educational requirements. Some areas have a strong preference to education, others a weak preference.
From what I have heard about in India, both through reading and through Indian co-workers, my understanding is that for the first few jobs the degree is critically important. Much like how India's social castes introduce social stratification, my understanding is that being able to attend college/university is heavily based on social strata. Since having a degree means you are likely in an upper strata, companies that employ software developers can use it as a way to filter out those of lower social status, employing only those they see as more desirable to have in their company.
Again from what I have learned but never experienced, in India the meritocracy of software development is not very strong. It is far more important to be from the upper social strata, to have the right family and social ties, than it is to have the skills. The skills are necessary, but that isn't what allows you to enter the workforce or drive your career forward. Those I know who came to the US told me how working overseas for a few years in part was their way of bypassing that system: they could not advance in their own country, but coming to the US and saving up as much as they could over that time would allow them to pursue better jobs when they returned.
