Cultural difference?

I listen to a short ESL radio program by American teachers every morning.
Today's skit was a conversation between those who don't know each other.
"Hi, it's cold today, isn't it?"
"Yeah, it's been cold."
"By the way, I'm Steve."
"I'm Jennie. Nice to meet you."
"Nice to meet you."

Is it truly natural in the US?
When those who don't know each other have a short conversation on the train or somewhere like that in Japan, we usually don't reveal our names. If someone who you don't know suddenly interoduces him/herself by name like in this skit, we would think he or she must be weird and beware of him or her.

Cultural Difference?

I listened to a short ESL radio program spoken by American teachers every morning.
Today's skit was a conversation between strangers those who don't know each other. [Nothing wrong with what you wrote.  However, why write longer when shorter ones suffice?]

"Hi, it's cold today, isn't it?"
"Yeah, it's been cold is/has." 
"By the way, I'm Steve."
"I'm Jennie. Nice to meet you."
"Nice to meet you too." [This is more polite, or more casually,  "nice meeting you [too]."  

The above conversation seemed a bit contrived. 

Is it truly natural in the US?

When those who don't know each other strangers engage have a short conversation in small talk on the train or somewhere like this in Japan, we don't usually tell announce our names [or "introduce ourselves"]. If someone who you don't know we suddenly blurt out our names to each other like this, tell his or her name like the skit, you would think he or she must be weird we will consider this as weird and beware him or her be wary of them.

Yes, it is a cultural thing.  It is considered polite if we introduce ourselves.  Remember, your best friend was once a stranger!  If you don't announce your names, it won't be considered as rude either. Also this shows interest in the other person.  Of course, if you consider the person whom you are talking with is handsome to you, you won't slap his face either!  You will be happy and glad!  If he is not your fancy, then you call him a weirdo or a creep.  Such is human nature!

I always introduce myself to strangers.  To me it is a courtesy and being friendly.  Of course how you do it depends on your tone and body language!

Cultural difference?

I listen to a short ESL radio program by American teachers every morning.
Today's skit was a conversation between those two people who don't know each other.
"Hi, it's cold today, isn't it?"
"Yeah, it's been cold."
"By the way, I'm Steve."
"I'm Jennie. Nice to meet you."
"Nice to meet you."

Is it truly natural in the US?
When those people who don't know each other have a short conversation on the train or somewhere like this in Japan, we they don't usually tell (=or give out) our their names. If someone who you don't know suddenly tells you his or her name like in the skit, you would think he or she must be weird and you would want to be beware of him or her.

Cultural difference?

I listen to a short ESL radio program by American teachers every morning.
Today's skit was a conversation between those who don't know each other.
"Hi, it's cold today, isn't it?"
"Yeah, it's been cold."
"By the way, I'm Steve."
"I'm Jennie. Nice to meet you."
"Nice to meet you."

Is it truly natural in the US?
When those who don't know each other have a short conversation on the train or somewhere like this? In Japan, we don't usually tell our names. If someone who you don't know suddenly tell you his or her name like in the skit, you would think he or she must be weird and beware him or her.

So now to your question,

I have been living in America for 8 years an I noticed that it is much more acceptable to make small talks and introduce yourself to others over here than in other countries (I come from Israel, which it is usually not acceptable).

However, it really depends on where you are in America. People in the countryside and in small towns or in the suburbs are usually much more friendly than city people. People who come from small town or the countryside will usually be friendly and talk to you more. However, in big crowded cities where people feel less secure and more rushed (like in New York City), people are less likely to talk to people they don't know out of the blue.

It is not a clear cut answer and in both cities and small towns, you could find all kinds of people but what I have said is generally true. Either way, that's an ESL radio program and the skit was staged in a way to teach you to introduce yourself :) I hope that helps.