In American football (NFL), is it possible for both teams to challenge on the same thing for the same play?
No. Once one team challenges the play, that challenge covers all reviewable aspects of the play, at least as of the most recent rules (2015). From Rule 15, Section 2, Article 3:
Note 2: All reviewable aspects of the play may be examined and are subject to reversal, even if not identified in a coach’s challenge or if not the specific reason for a Replay Official’s request for review.
Thus, one review covers all possible reviewable actions. A second flag is not explicitly discussed explicitly; however, it would be unlikely that a referee would allow such a challenge (and if he did, it would almost certainly just be a wasted timeout).