The reason every element has a unique atomic number and a unique symbol is based on the organization of the periodic table and the principles of atomic structure.
The atomic number of an element corresponds to the number of protons in the nucleus of its atoms. Since each element has a unique number of protons, it also has a unique atomic number. This fundamental property of elements allows them to be arranged in ascending order of atomic number in the periodic table.
The symbols of elements are derived from their names and serve as shorthand representations. The symbols are usually one or two letters derived from the element's name in English or another language. These symbols are standardized and provide a concise way to represent each element.
As for hydrogen, it has a unique situation due to historical reasons and isotopes. Hydrogen commonly exists as three isotopes: protium (with a single proton and no neutrons), deuterium (with a proton and a neutron), and tritium (with a proton and two neutrons). The isotopes of hydrogen have different masses but share the same atomic number, which is 1.
To distinguish between these isotopes, the IUPAC (International Union of Pure and Applied Chemistry) has approved the use of different symbols. The symbol "H" represents the element hydrogen in general, while "1-H" specifically refers to protium, "2-H" refers to deuterium, and "3-H" refers to tritium.
The use of multiple symbols for hydrogen isotopes is an exception to the general practice of using a single symbol for each element. It is done to accurately represent the different isotopes and their distinct properties.