Confusion regarding registering for two board exams and clearing one

While going through your submission, it appears that you have passed your tenth board from CBSC and later you opted for Bihar Board for clearance of class twelve in 2021 and finally you got through this examination.
Please note that your passing out two different board examinations are not likely to create any impediment in relation to your further studies since both the boards are well recognised and affiliated to the different institutions.
Hence concentrate on your ongoing studies without any unnecessary stress to achieve your goal.

As mentioned above, you passed 10th exam from CBSE and when you got fail in class 12th in CBSE then applied for Bihar board for class 12th and finally passed class 12th from bihar board school examination. Both board are recognised so there would not be any problem in taking further degree. So, you should stop unnecessary thinking and concentrate on your studies.

After doing your class 10 you did your class 12 from CBSE but you failed in that. It seems that you applied for class 12 exam of Bihar board and also applied for CBSE re exam in 2021. Probably you thought that you will use the credentials of the exam in whichever you fared well.
Actually when you applied for Bihar board the you should have forgotten about the CBSE re exam. That was an unnecessary step you had taken. It is good that you did not attempt that due to health reasons and could not appear in that exam. Now this is not a bad situation as you have not given the CBSE re exam and have already passed in Bihar board class 12 exam.
You have to mention about your failed attempt in CBSE class 12 and then mention the pass result of your Bihar board. That completes your biodata and there is nothing wrong in that as after a fail attempt you have a pass result. Forget about the re exam thing as it did not materialise and gave any result in your favour or against. There is no need to mention it even.
So prepare your biodata accordingly and present it with all the supporting document.

Nothing to worry about. You have not done any mistakes.
1. You passes your 10th class of CBSE in 2018,
2. You passed your 11th class of CBSE in 2019.
3. You could not clear your 12th class in the year 2020.
4. Then you have changed the board and opted for Bihar secondary education board. Appeared for the same and passed with a good percentage.
5. You have not appeared for the 12th of CBSE again due to health problems.
Actually, you need not mention about your CBSE second attempt as you have not attended the same. Leave that at that place and you need not mention the same anywhere. But you have to show that you failed in CBSE and then attended the board exam.
Once you make your bio data as above you will not have any problems in future while you apply for jobs. Concentrate on your.