This does seem to be a bit unspecified. Though there are rules for things that appear to be similar on the surface.
One of the similar rules is Redundant Conditions
Any ability that removes a condition removes it entirely, no matter what its condition value is or how many times you've been affected by it.
But a disease in an affliction. An affliction is not a condition.
Another similar rule is Flat Checks
If more than one flat check would ever cause or prevent the same thing, just roll once and use the highest DC.
But Treat Disease and affliction saves are not flat checks.
However, those rules do give some hints for how to handle this case. I am seeing two options. There may be more.
1) As you mentioned, use Treat Disease once and use that roll against all diseases that the target is afflicted by. Any that it succeeds against would get the bonus to its next save.
2) Use Treat Disease separately for each disease that the target has. Because of the time needed for this (8 hours) it would likely need to be done by separate people so that it could be done concurrently rather than sequentially - 16 hours is more than the 10 hour onset period. A modification to this ruling could be made to allow one character to treat all diseases at the same time, but still roll separately against each one.
Each of these options has good points and bad points to them. I'm not finding any rules to concretely say that one is more right than the other, so I think it is a matter of preference.
In any case, I would think that the party can only Treat Disease against diseases that they are aware of.