The Chabad-Lubavitch worldview regarding the universality of The Lubavitcher Rebbe is based off the belief that the Lubavitcher rebbe is the "Nasi Hador" the leader of the generation.
Chabad.org explains this concept:
Neshamah Kelalit; Nasi HaDor: the comprehensive soul of a Rebbe which
is bonded with the souls of all the Jews of his generation
As well as Jewish Virtual Library:
The Lubavitch Rebbe, as Nasi HaDor (leader of the generation) has the
responsibility of setting the direction of the generation.
As well as this site:
Lubavitch claims that the Rebbe is Moshiach because he is the Nasi
HaDor and the Rebbe said that the Nasi HaDor, the leader of the
generation is Moshiach.... Lubavitch [however], certainly believes that the Rebbe is still the Nasi HaDor. No matter who you’ll ask in Lubavitch, even the most vocal person who speaks out against saying the Rebbe is Moshiach - [if] you’ll ask him, "Who do you believe is your Nasi, is your leader and the leader of this generation?" he will certainly say that it’s the Rebbe. And that belief is the universal belief of all Lubavitch and many people outside of Lubavitch, who daven Nusach Ashkenaz, who follow Non-Chabad minhagim, who do not consider themselves to be Lubavitch. But many of them are afraid to come out openly, and to say that that’s what their belief is. Although some of them have
They believe that he inherited this title from the Sixth Lubavitcher Rebbe. As the seventh Lubavitcher Rebbe constantly referred to his father-in-law as the Nasi Hador (even post death).
As told to me by many Lubavitcher Shluchim-emissaries, the basis for this belief is that the Lubavitcher Rebbe was at one point the most recognized Rabbinical figure in the world, his opinions were sought after by politicians, gentiles and other Rabbinic figures. To a certain degree his opinions were taken as a legitimate representation of Judaism. And as Nasi Hador, all of the Jewish peoples souls are included within his soul, this is known as "Neshama Haklalis". Therefore, every Jew and his soul are intertwined and tied up with the soul of the Nasi Hador even if they are unaware of this. Thus, Chabad Chassidim believe that the Lubavitcher is the Rebbe and leader for everyone.
The implications of this belief are twofold. Firstly, from the Lubavitch perspective. Non acceptance of this belief from other non Lubavitch Jews is viewed with astonishment, they ask questions such as "how could you not follow his every directive? He is the Nasi Hador!?". However, from the non-Lubavitch perspective, this claim to title is very hard to understand, for (excluding those that follow the opinions of Maran Rav Schach) they look at the Lubavitcher Rebbe as a man of greatness who did many good things for the Jewish people and observance of mitzvos. But that doesn't obligate one to profess belief in assigning an official title of supremacy to him.
Furthermore, some Jews have a hard time understanding why the Lubavitcher Rebbe cannot be viewed within the whole of the rest of the Rabbinical Intelligentsia, rather than needing to elevate him above and beyond other Rabbinical figures.