Should the young man return to his old school to study a major he doesn't desire, just to be with his friends, or should he remain in the new school despite feeling boredom and depression due to his inability to adapt and the lack of close friends?

Study time is for seriousness and diligence, not for amusement and play. A beneficial friend is one who invites to good. As in the Almighty's saying: "And keep yourself patient with those who call upon their Lord morning and evening, seeking His countenance. And let not your eyes pass beyond them, desiring the adornments of the worldly life. And do not obey one whose heart We have made heedless of Our remembrance and who follows his own desire and whose affair has been a [matter of] neglect." Friends are gained through good character, good deeds, and kind words. It is advisable to remain in the new school with the desired specialization, provided it is not unlawful, and that personal desire does not affect one's best interest, for Allah has condemned those who follow their desires. As He the Almighty said: "but he clung to the earth and followed his own desire. So his example is like that of the dog: if you chase him, he pants, or if you leave him, he pants," and His saying: "And who is more astray than one who follows his own desire without guidance from Allah?" To treat depression, boredom, laziness, and lethargy, you can refer to relevant fatwas.