Ruthi is right. There is also the larger question of East-West differences in concepts of privacy. Age is regarded as private information (like salary) in many Western countries, except when one is a politician or a director of a publicly listed company.

Maybe because in the USA young people are more popular in society than older people. The culture glorifies youth and young people and if you are an older person you don't want other people to know your exact age, and some people prefer to look younger than their real age.

It's not a big deal, and it all depends on the situation. In the United States, as a rough rule, if you know somebody well enough to know when their birthday is, you know them well enough to ask their age. You would not ask it of a stranger. It is not a serious insult, it is just considered "personal." There is a traditional cultural bias in the United States that youth is good and age is bad, How strong it is depends very much on the time and place. It is stronger for women then for men. It has always been true. When I was a child in the 1950s, some women, if asked their age, would routinely say "over 21" and no more. It is not an absolute thing. Yes, in the United States we, too, admire our ancestors and respect the wisdom of elders--to some extent. But it is less strong than in Asian cultures. I was a software engineer. That's a job in which it is not too important how you look or dress. You don't want to offend your fellow workers, but you are not trying to look good for customers. My hair began to turn grey in my late fifties. I never considered dyeing it. However, in jobs where men are trying to look good, project confidence and power, it is quite common for men to dye their grey hair. (They even try to dye it just enough to look young, but not so suspiciously dark as to make it obvious that they are using dye!)

I don't know about other countries, but in Italy we are taught it's not polite to ask a woman her age. I guess it has to do with the expiring date a woman is put on by society, and being asked your age is often the first step in a sequence of judgments on you being successful by social standards put on women (looking beautiful at all times and ages, marrying very soon, having kids and very soon), all things a person usually has rarely control of (time, being liked by someone, etc.). You can see why some people would push for you to not talk about that, ever. Or it's some gentleman code for guys who are supposed to be charming and a peaceful distraction for women, so bringing up the negativity associated with the whole time thing would be against that.