Is it possible for someone to attain the same rank as Abu Bakr Al-Siddiq in Paradise, or is that a rank exclusive to him and the Companions?

No one after the Companions, generally, and Abu Bakr specifically, can attain their rank in Paradise, due to their unique virtue of accompanying the Prophet, peace and blessings be upon him. Even if someone after them were to spend gold equivalent to Mount Uhud, they would not reach the measure of what one of them spent. Many hadiths have been narrated regarding their superiority, for they are the best of people; Allah chose them above jinn and mankind, with the exception of the prophets and messengers.