Nicknames are a form of pseudonym and may or may not include "the" in the nickname, it depends what is intended by the nickname, also a nickname does not need to be a single word. but usually has some context or reason for a particular nickname. For example, in many situations one's nickname becomes the name one is socially known by
and one's friends would simply use it as they would a given name
Ender, do you want to join us?
Hey Top Drawer, what are you drinking tonight?
In UK public school, traditionally one is addressed by one's surname as a form of nickname.
Using "the" would be a designation to make the nickname more of a title
He's known as The Ender.
"The Donald" is the new president elect in the US.
However, people would not usually use "The Ender" to directly address you, just as they would not use "The Alex", when speaking to you.
In your example, if you wrote your name as
Alex "The Ender" Gnatyuk
your nickname would be "The Ender", however if you wrote it
Alex "Ender" Gnatyuk
Edwin Eugene "Buzz" Aldrin Jr.
your nickname would be considers to be "Ender", and people might say
That guy is "Ender"
That guy is "Ender" Gnatyuk.