How is it permissible to eat the slaughtered animal of the People of the Book that was not consecrated with the mention of God’s name at the time of slaughter, and does this include the Christians and Jews of the present age, whether those residing in Muslim countries or outside them?

Yes, the Jews and Christians of today fall under the designation "People of the Book." Allah has made lawful their slaughtered animals and marriage to their women for Muslims. Their disbelief, polytheism, and alteration of their scriptures did not prevent these rulings from being applied to them during the time of the Prophet, peace and blessings be upon him. Therefore, the rulings concerning the People of the Book remain applicable to them until the Day of Judgment.