SQL Server: Backup failed because there is a mismatch in file metadata
When we do a DBCC, no errors get reported, DBCC CHECKDB ('MyDB', repair_allow_data_loss) also does not return any issues

This is really interesting as DBCC should report errors if metadata is corrupted. Have you tried to run a DBCC CHECKCATALOG; ?

Backup failed because there is a mismatch in file metadata for file 65537

I know it sounds basic but have you checked what database does file 65537 belongs to? Make sure the DBCC CHECKDB and DBCC CHECKCATALOG are run against the good database. But judging from the fact that you have problems with your backups, I guess you are looking at the right database. Unless it would be msdb? Since backup sets are in msdb so when you do a backup it inserts rows in msdb also.
You should check Paul Randall's blog, he has a post on fixing metadata corruption without a backup :
https://www.sqlskills.com/blogs/paul/disaster-recovery-101-fixing-metadata-corruption-without-a-backup/
Did you try to restore your backups to see if they are corrupted?
Do you have filestream enabled? From what I'm reading, file 65537 would be related to filestream.
https://www.sqlshack.com/sql-server-filestream-database-corruption-and-remediation/
There's also a great blog by Paul Randal on filestream you should check it out.
https://www.sqlskills.com/blogs/paul/filestream-directory-structure/
-------------- EDITED for FileStream corruption -------------------------
You should check the answer on this question that explains what happens when Filestream is corrupted.
https://stackoverflow.com/questions/15302673/handle-sql-filestream-data-corruption-and-backup
And in the link on sqlshack, there are steps to recover a database from filestream corruption.