The Teamwatch ability applies to fielding a different character because the definition of Teamwatch states “another character” and such words have been ruled to mean a different character.
From the Age of Ultron Rule Book, Lexicon:
Teamwatch: Teamwatch abilities happen when a character with Teamwatch is active, and you field another character who has the same affiliation as them.
There is also a FAQ that generally covers this game effect as well regarding the Teamwork basic action card:
FAQ 20 (22Sep2014)
Q: How does the Teamwork effect work?
A: If you had an Iron Man, two Thor, and a Hulk in play, each of them would get +2A and +2D. Teamwork identifies that you have 3 different Avengers in play and it doesn’t matter how many copies of those dice are in play. The bonus of Teamwork occurs when played, and doesn’t change if a different number of characters are fielded after it is used.
So in your example, if Hawkeye - Clint is active, anytime you field another Avenger, other than Hawkeye, his ability is triggered.
If Moondragon - Heather Douglas is active, she gets +2D for each Avenger/Guardian, other than Moondragon, that is fielded.
If both Hawkeye and Moondragon are active, you would trigger their applicable abilities but only if the fielded die is different than the ability triggered, i.e. if you field a Hawkeye, only Moondragon would trigger and visa-versa if you field a Moondragon.
Hawkeye - Clint
Teamwatch - When you field a character from the same team, deal 1 damage to an opposing character.
Moondragon - Heather Douglas
Teamwatch - when you field a character who shares an affiliation with Moondragon, she gets +2D this turn.