Does SQL Server full restore set the differential base?
The differential backup is always associated with the last full backup. It is that simple.

So any differential restore is looking for a certain full restore.

So yes, the base is reset. You can't mix and match.

Personally, I wouldn't have monthly full backups and hourly diffs. The frequencies don't match (example: weekly full, daily diff, hourly log). And I wouldn't use any binary file diff rubbish for my backups because I expect to be able to restore them.