Ubuntu – Why could I not export variables permanently

What is this "export" phrase at the start?

export is a command (more precisely it's a Bash builtin, i.e. it's not an executable present in PATH, it's a command that Bash has built-in in itself).

Is it exporting the data to be available for Bash?

export sets the environment variable on the left side of the assignment to the value on the right side of the assignment; such environment variable is visible to the process that sets it and to all the subprocesses spawned in the same environment, i.e. in this case to the Bash instance that sources ~/.profile and to all the subprocesses spawned in the same environment (which may include e.g. also other shells, which will in turn be able to access it).

What is the first PATH and what is the second $PATH, and why do we need two?

The first PATH as explained above is the environment variable to be set using export.

Since PATH normally contains something when ~/.profile is sourced (by default it contains /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games), simply setting PATH to ~/.composer/vendor/bin would make PATH contain only ~/.composer/vendor/bin.

So since references to a variable in a command are replaced with (or "expanded" to) the variable's value by Bash at the time of the command's evaluation, :$PATH is put at the end of the value to be assigned to PATH so that PATH ends up containing ~/.composer/vendor/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games (i.e. what PATH contains already plus ~/.composer/vendor/bin: at the start).