This is my sentence:
East and west meet each other at some points and even give hugs to each other.
I am not sure if this is a correct usage of meet and point to convey the meaning, which is:
East and west act/react the same about some special matters sometimes.
I searched for the exact sentence and didn’t find it, then I tried to look in dictionaries for the word point:
3 [countable] a particular moment in time
at this/that point:
At that point we all got up and walked out of the room.
at this/that point in time:
At this point in time we can’t afford to hire any more people.
4 [countable] a particular place
We’ll meet at a point halfway between here and your hotel.
A point where three different countries meet
Whenever you speak of East and West “meeting” you are echoing a once very famous poem by Rudyard Kipling, “The Ballad of East and West”, which begins and ends
OH, East is East and West is West, and never the twain shall meet,
Till Earth and Sky stand presently at God’s great Judgement Seat;
But there is neither East nor West, Border, nor Breed, nor Birth,
When two strong men stand face to face, tho’ they come from the ends of the earth!
Although it is unlikely that many of your readers will actually know the poem, the first line is frequently quoted and has become a generally known catchphrase. Here meet has just the meaning it seems you intend: to agree, to share a common outlook. We speak in the same sense of a meeting of minds.
Note that you don’t need the phrase each other—that is ‘built in’ to the verb.
Some points in the sense of some matters is idiomatic. (It’s meaning #1 in the dictionary you link to.) But at some points is used of physical or temporal locations, which is not what you want. (After all, geographical East and West do meet at infinite points along the border which divides them!) We speak of agreeing or disagreeing on points, rather than at them.
Give hugs to each other has a domestic, sentimental ring which I think is in the wrong register for what you are saying. I suggest embrace instead—which is doubly appropriate because we speak of embracing not only people but ideas and attitudes, too.
I think what you want is something more like:
East and West do meet, and even embrace, on some points.