Is Captain America captain of the Avengers team? I am saying this because Captain America takes most decisions and once in Avengers: Age of Ultron (2015). Tony Stark mentions that he is just the financier and designs everything, Captain is the owner.
Maria Hill: Lab's all set up, Boss.
Stark: Actually he's the boss [indicating Steve]. I just pay for everything and design everything, make everyone look cooler.
I would say that, if there had to be one, Cap would be the obvious choice. As others have mentioned, he has a history of leading others from his work before he was encapsulated in the ice. He also is very mission-oriented and very much interested in the greater good at his own personal cost. Whether this makes someone a good leader or not is debatable but when considering the goal of The Avengers they seem to be positives.
Why not the others?
Thor tends to wander off back to Asgard to take care of things there and it's difficult to lead a group when you have multiple priorities. He may have more battle experience but Earth is (arguably) secondary to Asgard.
Banner tends to wander off because he's not in control of the Hulk's power and doesn't want to hurt others, and lacks confidence. He's a great scientific mind but his heart isn't in the fighting.
Stark has more money than god and more brains than Einstein but he's very self-centered. Sometimes it takes a bit of prodding for him to do the "right" thing and his version of what's right isn't always the same as what's conventionally considered "right". Plus, as Fury was the one setting up The Avengers and intentionally tried to keep Stark out, he doesn't seem to be trusted as a leader.
Romanoff is used to working alone. She'll work as part of a team when necessary but her background is as an assassin/spy, a character type prone to working solo. And, yes, I'll play the misogyny card... she's also a girl. Girls don't generally get to lead men in superhero media.
Barton shows a lot of leadership qualities and the team does look to him often, which is shown pretty clearly in Age of Ultron. But he's also very modest and his family is clearly very important to him, so I think he'd avoid the position of leadership even if it was offered to him.
Plus, if you look at the end of Age of Ultron, at who's left for The Avengers team, it's a completely different group of people and the only two from those above are Cap and Romanoff.
Oh, and what about Fury?
Well, I'm pretty sure they make it clear a couple of times that, while SHIELD may have been behind creating the Avengers, they're not going to run the group. Fury has his own things to work on, managing what's left of SHIELD (as discussed in Agents of SHIELD). They'll be there for back-up if needed, as seen in Age of Ultron but they're two separate entities.
If by captain you mean "Leader" then yes. From all the Avengers Captain America has the most experience in combat and leading a team from his battles in WW2 (with the exception of Thor maybe). This qualifies him to lead the team because he's able to make quick decisions and form a battle plan while still being able to fight well without being too distracted.
In the MCU, Captain America is shown to be the leader. This is evident from the main battle in Avengers (2012), where Tony Stark tells Captain to make the battle plan.
To go off the scenes,
These points make most people think of him as the leader of the team.
Captain America is the leader for the Avengers. If we look back at the past movies for the Marvel Cinematic Universe, there are many points leading to this. Iron Man is taken off the team for the Avengers in Iron Man 2 after Black Widow's report on his life. When they later put him on the team in The Avengers, it is to look at Selvig's research at first, since he is just a consultant. He is later added to the team. Captain America was seen as an American hero in the 1940s for helping in WWII, and was known across the globe. He was seen as a symbol of hope for Americans, showing how strong we were against our enemies. Getting found in the ice probably sparked the public due to the fact that they had heard the stories of the American hero. S.H.I.E.L.D most likely put him as the leader of the team also because of this, as the public would be more open minded to the Avengers. Though Captain America is said by Tony in the Age of Ultron trailer to be the captain, there are many points that show this in the first place.
Yes. According to the official Marvel Studios: Character Encyclopedia, he is described as quite literally being the captain of the Avengers.
...He wakes up in the present day and is given charge over Nick Fury's Avengers Team.
As captain of the Avengers, Steve Rogers leads Thor, Iron Man, Hawkeye, Hulk and Black Widow. Occasionally he clashes with Thor and Iron Man.
