I would like to know how I can make my GNU/Linux GUI programs have a custom icon while running. I mean the icon that in some window managers appears at the top left corner of the window or to the left of the title in the task bar.

I've read many times that there isn't a "program icon" in GNU/Linux, but I know that something of the sort does exist because when I compile my programs, they show with a default icon and when I download some other program, it does have an icon. Also, I use FreeBasic and it does have an option to compile a xpm icon file into the binary, so I wonder how that is done internally and how I should do it with Flat Assembler

Just to get you started: To show an icon in Linux you have to communicate with the window manager in the desktop environment you run.

Since each DE/WM is different you will either need to write a few different versions of your executable, one for each case, or you can use one of the HLL "glue" libraries like GTK, QT, or similar.