I think you're asking two questions at once:
Do we need to break countries down into sub-tags for other areas of jurisdiction?
I don't have answer for that, and it seems like the community doesn't, either. There are tags for "California" and "Florida" but not "New York."
Second, assuming we justify that level of detail,
Is "Ontario" the right way to tag Ontario, Canada?
I think we have an answer to this when you said,
there are many places called Ontario, not just the one in Canada
If we decide to keep the tag, it should probably be edited into Ontario-Canada or something similar.