That is essentially correct, and we actually do call many different numbers "0" in different contexts, but it's not necessary for any two numbers to share a name this way. You can give them all unique finite names with an infinitely large alphabet if you prefer.
You might consider that in the existing Chinese writing system, 口 and 囗 are distinct symbols.
You might consider that in the existing Chinese writing system, 口 and 囗 are distinct symbols.