The question of why a city auditor would review a county budget is answered by a reader, MJC, who did some digging and emailed this:

The accounts of the trustees in each county except Suffolk shall be audited by the director of accounts and in Suffolk county shall be audited by the auditor of the city of Boston in the manner in which other county accounts are audited under general law.

So now the question is, Why are things different in Suffolk?