You're correct.
In very, very broad US-centric* strokes: Using a mark in trade is enough to establish a defensible trademark.
Registering a trademark can be useful, but it is also optional. At very least, registration helps make the ownership of the mark easier to discover and this can help everyone start on the right foot.
(* I'm not familiar at all with the laws of France, but that's fine: The alleged violation happened in New York.)
> In very, very broad US-centric* strokes: Using a mark in trade is enough to establish a defensible trademark.
Isn't that only if it's something that would actually qualify for a trademark?
For example, "Car Shop" or probably even "Hamburgers USA" would not qualify for a trademark due to being overly generic/descriptive (in many jurisdictions).
Now in Notepad++'s case the inclusion of the ++ obviously means it would indeed qualify.
Just asking as I'm sure there's people around here with personal experience around the topic, though again it can differ quite a bit by country.