Where does it imply or say in this verse that you have to make sure that they slaughter the animal according to their religion and pronounce the name of one God ?
Not in this ayah but in this "Surah Inam(121) Hence, eat not of that over which God's name has not been pronounced: for this would be sinful conduct indeed....."
In fact it does not mention slaughtered animal it gives an open license to eat from their food. Since eating pig is strictly and explicitly prohibited in other verses so I take that exception.
pick n choose ayahs....hmmmm.
If they believed in one God then there will be no segregation between Muslims and People of the book . Will they be not classified as Muslims ?
No, because after the Quran has come, muslims are those who believe in the Last Prophet(SAW), we still have Christians and Jews who believe in One God but they're called Unitarian Christians or Unitarian Jews.