Unfamiliar with the series I looked up a little more background information and found it on TVTropes, as an instance of the Don't Explain The Joke trope:
Finch poses as a blogger named Thomas Paine, which he admits is a pseudonym (his "nom de plume"). A few moments later, he offers someone a pen with his name and phone number on it and explains that it's his "Plume de nom..... rather than nom de plume."
So yes, your interpretation is right.
A nom de plume is a "pen name" or pseudonym. The plume - the feather originally used to write one's name in ink - in this case is a fountain pen which has the author's real name engraved on it. Hence they refer to it as their plume de nom ("feather/pen of the name"): not their 'name of writing' but their 'tool for writing their name'.
It doesn't make complete sense, since the pen can still be used as the plume to write their pen name with, and the pen itself has intrinsically nothing to do with the name engraved on it, convoluting the plume with the nom. But it sounds quite intellectual, and I guess that's what the writers went for.