In other words, mature is also well-known as a subject being fully-grown or developed, an adult. This is where you may ask yourself how is age does not relate to someone being mature, in this case, physically.
Though in this thread, you are talking about the process of maturity, which means that the subject may or may not have achieved it. And I'd like to believe that you all are talking about it in the mental, or emotional way. You may experience and learn it from your life journey, and assume that you are mature, or not. However, this is where most people thought wrong:
"I may be young but I have more experience than older people."
It's just a form of a dialog I hear a lot from people nowadays. How much do you think is a lot of experience? What if your experiences are just the fraction of what others experience daily? A lot of people now take that as an excuse to say that they know what they are doing. But take this as another process of maturity. You eventually learn from your actions itself, and it becomes your experience in the future, you will remember it again and knock your head on the wall because you will cringe at yourself.
Personally, I like to put "mature" and "moderate" side by side. They would know how much is too much, how much would be of enough quality to them or to the others. This would usually take a lot of consideration and you would know from the result of their decision, but as usual, we can't satisfy everyone. Sacrifice in some cases would be necessary, and you're willing to live with the consequences. Unless if you're smart, you could be creative enough to kill two birds with one stone. See, there are the words "smart" and "creative" there. I'd like to believe that mature people aren't necessarily smart, creative etc, but possessed a common sense (because seriously, nowadays common sense is so rare you'd consider it a superpower) to get around life.
Is there a specific definition of maturity?
Well check the dictionary, for a start. Or the internet. You can always ask people too, answers will always differ.
Do we have to reach a certain age to be labeled as mature?
Might be, physically, like I said. But mentally, it's up to an individual to decide it, it's the same as how they would define maturity by their own dictionary.
Do you see yourself as a mature person?
This is up to people, who knows if they think I am, or I am not. But whatever their decision is, know that I do not care. But from what I hear from people, on first impression they think I'm immature, but those who know me well say that I'm mature. People are very judgemental even when they don't know you, and tend to make up the impression of how a person is, until they would change their mind when they try to know you well. That would probably be one of the aspect of "immaturity"; too quick to judge.