Does my name have to be the same on all social media?
Yes, and here is why.
The name of your brand, your name, or company name is always the same, regardless of the language. If you had a different name each time you introduced yourself, it would make things unnecessarily complicated and would confuse a lot of people.
Recognizability of your account also plays an important part. By having the same name everywhere, you will make it easier for everyone to find you and remember you.
In case the name you want is already taken, try to add a certain character which will not change the meaning of the original name too much.