Is there a connection between frodo and sauron?

[Secret Fire]

I think the term we've all been dancing around is foil. A character's foil is another character who demonstrates similar actions/thoughts, or are put under similar circumstances.

Sorry Lindril, a foil is not a character who has actions/thoughts similar to another, but one whose actions/thoughts are opposite another character's actions/motivation/thoughts. The example you gave that I can see coming into effect here is Sarmon/Gandalf.
Dictionary.com defines a foil as One that by contrast underscores or enhances the distinctive characteristics of another: “I am resolved my husband shall not be a rival, but a foil to me." Another foil apparent, however, could be Denethor/Theoden.