Is someone who unbuttons the first and second buttons of his shirt considered to be of the people of Lut, and what is the ruling on this action; is it permissible?

There is no harm for a man in unbuttoning the first or second button of his shirt as long as it does not lead to the exposure of his 'awrah (private parts). Abu Dawud narrated in his Sunan from Mu'awiyah ibn Qurrah, from his father, who said: "I came to the Messenger of Allah, peace and blessings be upon him, with a group of people from Muzaynah, and we pledged allegiance to him, and indeed, his shirt was unbuttoned." This hadith was authenticated by Al-Albani.

As for what has been reported that this is an act of the people of Lut, that is a fabricated and false hadith. If there were any dislikeability in it, it would perhaps be if it led to the exposure of the 'awrah.

As for the imitation of disbelievers, which is blameworthy in Islamic law, it refers to that which is characteristic of disbelievers, not something that has become widespread among Muslims and no longer distinguishes disbelievers from others.