Q:

What is the basis for the focus on the difference between Jews and everybody else?

I have noticed that in a large portion of the questions and answers here, the distinction between Jews and other people is repeatedly emphasized.  I know enough about the Tanakh to understand that Judaism is a very unique covenant between G-d and His chosen people, but the degree of emphasis is surprising to me.  
Case in point: There is a word that, according to the glossary, is specifically used when comparing Judaism to other religions, Jews to other people, the Torah to other books, etc. - "L'vhadil!

The answer is that this is one of the differences between a Jew and l'havdil a gentile.
  -  Rabbi Shlomo Price

And:

“Honest politician” is an oxymoron, from the local level all the way up to Congress and the White House it’s the same story, Republican or Democrat, Jew or l’havdil Gentile...
  -  Source

The question isn't specific to l'havdil though, it is about the general phenomenon.  
Another example:  
Is there any obligation for a Jew to save a gentile's life?
And the answer to the question of  Are you allowed to save a non-Jew's life on Shabbos? is "Yes, but mostly because there would be backlash if you didn't".  
Instead of taking the position that life is life, and identity is secondary, the emphasis here is on whether or not the life belongs to a Jew.  
Why is there such an intense focus on the difference between Jews, Judaism, Hebrew Scriptures, and everyone and everything else?

A:

As many others have mentioned before, when creating laws, you HAVE to make certain distinctions. One of the biggest blindspots i've noticed with Christians trying to understand Judaism, is that they view everything from a theological perspective, and forget one very important point. Modern Christians are used to living in a society where the government is run by laws created by men, and the laws are changed, added to, and developed solely on what new situations dictate. Whereas in the times of the Talmud, or in the scope of the Biblical text, the Torah is not only the basis for religion, it's the basis for how you should run an entire country. And since the holy is mixed with the mundane, you will find distinctions between priest and normal israelite in the law, distinctions between the convert and the normal israelite (converts would not have an inheritance to the land of Israel like the 12 tribes would), and the list goes on. Just like in the United States, we have certain laws for citizens and non citizens. Non citizens are not allowed to go to certain schools, or have drivers licenses, etc. It's (hopefully) not because we feel we are superior to them, but that at some point we need to have some distinguishments.
For the Torah, this is also evident, and it's easy to focus on the abundance of scriptures for how Holy we as children of Israel are/should be. We are supposed to be a nation of priests, we are God's chosen, we are "His Children."
But i think sometimes we forget some other scriptures, the ones that are a response to those who think we are better than other nations, or holier, or that God has only done wondrous things for us. Scriptures that should humble us and help us remember, that at the end of the day, we all stand before the same Judge, and He doesn't pick favorites.

Amos 9:7-8
הֲלוֹא כִבְנֵי כֻשִׁיִּים אַתֶּם לִי בְּנֵי יִשְׂרָאֵל, נְאֻם-יְהוָה:
הֲלוֹא אֶת-יִשְׂרָאֵל, הֶעֱלֵיתִי מֵאֶרֶץ מִצְרַיִם, וּפְלִשְׁתִּיִּים
מִכַּפְתּוֹר, וַאֲרָם מִקִּיר.
הִנֵּה עֵינֵי אֲדֹנָי יְהוִה, בַּמַּמְלָכָה הַחַטָּאָה, וְהִשְׁמַדְתִּי אֹתָהּ, מֵעַל פְּנֵי הָאֲדָמָה:  אֶפֶס, כִּי לֹא הַשְׁמֵיד אַשְׁמִיד אֶת-בֵּית יַעֲקֹב--נְאֻם-יְהוָה.
Are ye not as the children of the Ethiopians unto Me, O children of
Israel? saith the LORD. Have not I brought up Israel out of the land
of Egypt, and the Philistines from Caphtor, and Aram from Kir?
Behold, the eyes of the Lord GOD are upon the sinful kingdom [Israel], and I
will destroy it from off the face of the earth; saving that I will not
utterly destroy the house of Jacob, saith the LORD.

A:

As has been discussed in the comments on the question, I think your understanding of the word "l'havdil" is not quite accurate. "L'havdil" really just means a distinction between things that are comparable in a particular sense, but not in a general sense (because one of the items of comparison is "holier" for some definition of that word). For example, I could say, "The Torah is written in Hebrew, and so is (l'havdil) the Hebrew translation of Harry Potter."
Your main question is about why there is a focus on the difference between Jews and everybody else (on this site and, perhaps, in general). The basic reason for that is simply that Jews and non-Jews are in different halachic categories. As such, the laws that apply to individuals are different for Jews and for non-Jews. 
This site in particular deals extensively with halacha. As such, it is frequently important to distinguish between Jews and gentiles because the halacha is different. Jews have responsibilities that non-Jews do not have. In addition, sometimes a Jew's responsibilities are different depending on whether he lives in an area that is predominantly Jewish or otherwise.  We focus on the difference when it is relevant. 

