If you see it in the Finder, then you should see it in the Terminal. The Terminal can always show more than the Finder.
I don't know what you mean about "basic names" and "nick names". In Linux/Unix, there exists a UID (User ID) that identifies each user. The root (super user) gets UID 0. Everyone else gets higher numbers. Normal user accounts on a Mac start at UID 501. Your user ID is probably 501.
But people rarely use User IDs. It is more common to reference a "user name", also called a "login name" or sometimes just a "login". Apple calls this a "short user name", because Apple always has to "think different".
Typically a person's login is the first letter of their first name and then their last name. "Sara Lee" would have a login of "slee". Again, Apple needs to just slightly different so Apple defaults this value to your first and last name concatenated together. If "Sara Lee" created an account on a Mac and used the default value, her login would be "saralee". (Old-school Unix greybeards have big hangups on this. None of these values would be acceptable for them. The only valid user login has 3 letters for the first letters of the user's first, middle, and last name. Anything more than 3 letters, for anything really, is an abomination for these folks. Assuming Sara Lee's middle name is "Ann", and she worked on a PDP-11 in 1981, her login would then be "sal".)
Apple also has a concept of a "User Name" as opposed to the "Short User Name". Any of the dialog boxes that pop-up asking for your authentication will default to, and prefer, your "User Name". As in the example above, Sara Lee's "user name" would be "Sara Lee". While this concept also exists in Linux/Unix, it is never used there. The only place it is used on the Mac is on those pop-up authentication dialogs.
The only important value here is the "short user name" which would be "saralee" by default. If you are doing really, really advanced stuff, then you might use that numerical UID value which defaults to 501. Ignore anything else.
Considering that you have already installed homebrew and allowed it to take liberties with your account, there is no way to tell what it has done or changed. It could have changed your Terminal prompt to read something strange which you are interpreting as "basic name" and "nickname". I can't tell you any more than that. Even on Linux, those package manager tools are vile abominations against nature. They are suitable only for short tutorials. Afterwards, they will then wreck your system for years to come.