Degree certificate issues in universities

In case of your friend, she has cleared her final B.Tech ( Final) in the year 2020 but due to her health issue, she could not clear the backlog pending for Mathematics. Now she has cleared the same. In her case, she is to be treated as par with the other candidates who have cleared the final examination on the date of publication of the result.
Owing to her backlog paper for which even she has successfully passed is not likely to create any obstacles other than she will have an additional mark sheet for the backlog where there will be indication of month and year of her appearance for the said backlog examination.
If the date for such an examination has extended beyond the date of publication of the final year results of 2020, it might affect the time of completion of your final degree.

Universities have got set rules in this regard and you will have to find out about this from their administrative office. Actually the backlog is not considered in the same year because it is not like the supplementary exams which we used to have long time back and were held immediately after the main exams.
Backlog exams can be after a long gap also. In fact the universities give an ample time to the tune of 2 years for this. So, a 4 year course could be completed even in 6 years. Depending upon the time of the backlog exam and its result announcement it ca be included in a particular year. I do not think that you can insist them to include the said one in the current year.

A student should clear all his backlogs before passing the final year examinations. By the date on which the final year results are declared if there is a backlog, they can't give you the pass certificate in that year itself. They will change the year. In your friend's case, she cleared the backlog after she passed the final year examinations. So no rule will say that the certificate should be given without changing the year.
But you can take a provisional certificate for the qualification and with that, you can apply for jobs or higher studies. The UGC will have rules and regulations and all the universities that are having UGC recognition should follow the same. Otherwise, their recognition will get cancelled. So they can't issue the certificate at their discretion. No law can insist the university to violate the rules that are prescribed by UGC.

For such issues you can approach directly University people along with the documents such as
1.Your College ID
2. Your aadhaar card for proof of your identity
3.Your earlier certificates of passing obtained
4. Hall ticket of the December 2020 appearance
5. document related to your passing the paper in December 2020 examination.

You can ask a consolidated Mark sheet for the degree from the university.
With that you can apply for Degree in absentia in the prescribed form available in the university office
But for the above everything your personal approach is necessary and no correspondence will do your favor.