In my experience, people who are well trained in mathematics are resilient and have confidence in mastering their daily life activities and solving their problems.
Is learning mathematics a good way of building personal character?
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I hate this answer, but: it depends.
Mathematics alone won't do anything. In your observation you are correlating mathematics skills with personal behaviour. What you can't see is whether math made them who they are or who they are makes it easy for them to understand math.
Focusing on becoming very good at something is character building. Encouraging your child to become top at something and going down a (means: precisely one, even in this figure of speech) road with your kid for years will definetly be character building. Yet, it doesn't need to be math.
However, a lot of our lives comes down to math. From telling time, buying gum with pocket money to making and understanding long term desicions and developments. Understanding exponential growth can unravel a lot of secrets of our everyday life.
And of course you are always better off if you understand that you are spending monney even when the add tells you you are saving a buck or two ...
If you replace "mathematics" with "hard science" or "engineering" then I think I see where you're going with this. Much like the first answer, I believe that logical thinking and a good understanding of how things work make it easier to be organized -- but it doesn't help with personal character per se.
You could be a math genius, or a fantastic engineer, with a precise understanding of the physical world, and still be a social weakling (or worse). You could have a deep understanding of the psychological and social aspects in our world, and still have a bad character.
What builds character, then?
Some would say survival training. Some say religious activities. Some say team sports. Choose whatever works for you; something you can see yourself doing for a long time.
I think the common denominator is to discover your personal strengths and weaknesses, learn how to make best use of them for the greater good of the community -- and then habitually exercise these skills.
About personal character not so much, learning maths won't change who you are. However it will help with logical thinking and problem solving throughout your live, thus it will help with confidence in solving tasks/activities.
The good thing about math is passion. With my experience math helped me in this way. Having a passion to solve something gives you the ability to keep on trying no matter how long it takes, there has to be a solution. How I use this in real life is that no matter what rock life may throw at me, there will always be a way, there will always be a sun shining at the end of the road when I figured all the solutions out and done what I had to do to get there.
So no, math can't be used to build your character, but you can use the tactics you use in math in your life. If done with passion and willingly then yes, you may be able to add something to your character (who you are and not about what you are doing e.g. determined).
Hope I helped a bit....
I think the causation is flipped. In my experience, getting very good at anything, but especially at something abstract like math, is difficult. People who overcome initial difficulties to become good at something generally do so because they have a strong character. A few do so because their parents essentially force them to (music lessons come to mind), but typically, they later don't mention or show those abilities, preventing you from correlating the abilities with their character.
What's more, having lots of strength and confidence is not the same as having a good character. Some adults use their math skills, their speaking skills, and their confidence to hurt others by taking unfair advantage of them and cheating them. I wouldn't call that having a good character.
Math is useful. Encourage your children to learn it: I did. But you teach them how to have a strong character and be a good person by teaching them character and goodness, not teaching them math and hoping it will magically make them better at those things. Working towards improvement at anything (math, music, figure skating, cooking, painting, science, writing, poetry...) will teach the values of perserverance, patience, and consistency. Being pushed into working at something you don't like and aren't good at, especially when the work is not its own reward but is something to make you stronger later, is something I would not have wanted done to me. I think it might even backfire.
I don't know how much it builds character, but it provides one with an alternative (and perhaps even refined) way of thinking about problems (and perhaps even, the world).
It never hurts to have different ways of thinking about problems.
I would also like to add that learning how to write computer programs provides one with another way of thinking about the world, which is different from mathematics.
In my experience, it helps that a person will avoid awkwards and potentially embarrassing situations such as visibly trying to or having trouble adding or subtracting in a daily experiences.
If they have math under the knee (they know it hundred percent) they won't feel uncomfortable or worse, going around with a feeling of inferiority. Every time they do need to do math, their minds freezes, because they think they don't know it, etc.