Does my name have to be the same on all social networks?
Yes, and here's why.
Your brand name, personal name, or company name is always the same, regardless of the language in which you pronounce it. If you had a different name every time you introduced yourself, it would further complicate things and confuse everyone around you.
The recognizability and memory of your account also play an important role. By having the same account name, you make it easier for everyone to find and remember you.
In case the name you want is busy, try adding a character that will not distort the name you originally wanted.