Fine with me, QC-passed. I would just change it a bit, in the first line.
Yours: My friends always call me by my nickname.
Mine: My friends always call me by the nickname.
This bit also could be changed:
I always have to explain to him/her that he/she can call me by my nickname but that's not my real name
I always have to explain to him/her that he/she can use the/my nickname [can use any other name] to call me. Only/except that it´s not my real name.
1) I think one presumes the nickname is yours, they certainly wouldn´t call you by the nickname of somebody else.
2) When you use he/she to make it inpersonal, you can use "they", for example: They think my nickname is the real/my true name. Anyway, I will leave that bit of grammar to Adonay, Tradutora, Telma, Thomas and others to elaborate more.
These are just some advice to let you know that you can change things a tad, so your English gets a bit more colourful and natural. And less repetitive in the process. I mean, you don´t have to repeat "by my nickname" twice in a piece of text, if you don´t wish to, of course.
It was supposed to be just the first line, but I kept going...excuse me!