They use their aliases due to the fact that the Matrix is a prison and they were born into bondage and given that name. Their aliases on the other hand are created by their own mind and it is personal to them.
Not only that but the names are all relevant to hacker terms, phrases etc. Which is in fact, what they do. They were no doubt hackers in the Matrix and they now hack into the Matrix.
Then we have the reason behind their name, it's purpose. They all have a purpose. eg. Neo = Revival of something old, in a new form (something like that) or New (Neo/New Tokyo). Neo is also an anagram for One which we all should know. But could this possibly be a phrase?
New One - hrmmmm. Interesting 😉
So, they use their hacker aliases because of the above reasons and probably more.
It's a kind of reversal of the Prisoner thing (you know, that TV show filmed at Porthmeirion) - instead of 'not being a number'... they are a number, well, an alias. Actually, they do kind-of have numbers... Neo's is something like TA042SS62... see the yellow room scene in the original Matrix when Thomas is bugged.
~AI