The age of the Earth and the age of the universe are indeed different, but they are not directly related in the way you described. While the age of the universe is estimated to be around 13.8 billion years, the age of the Earth is estimated to be about 4.6 billion years.
The concept of "primordial soup" generally refers to the hypothetical mixture of organic compounds that could have existed on Earth's early oceans, which eventually led to the development of life. These organic compounds are thought to have formed through various processes, including chemical reactions involving elements such as carbon, hydrogen, nitrogen, and oxygen.
The elements present in the universe, including those found on Earth, were synthesized through stellar nucleosynthesis. This means that elements heavier than hydrogen and helium, such as carbon, oxygen, and iron, were formed through nuclear reactions within stars over billions of years. When certain stars reach the end of their life cycles, they explode in supernovae, dispersing their enriched material into space. This material then mixes with other interstellar gas and dust, eventually forming new stars and planetary systems.
Therefore, while the atoms that make up Earth can be traced back to processes that occurred over billions of years in the universe, it would not be accurate to say that the Earth itself is 13 billion years old. The age of the Earth refers specifically to the time since its formation as a distinct celestial body within the solar system.