I agree it is Eu-wide or pan-EU rather than pan European.
Its probably not going to solve the problems it sets out to given all the differences between EU countries legal systems, tax, regulation etc.
That said I think the headline proposal (single entity type and single registrar) is not important. The UK and (AFAIK) the US have few practical difficulties with multiple registrars and variations between jurisdictions.
And by "it" we mean both "free choice to be part of the union or not" and "legal jurisdictions".
Much like "an American firm" doesn't mean Canadian or Brazilian.
See comment above:
> Its obviously EU
I know how continents work. I don't think you know when contextual usages of language work.
One might have said the use of the word "American" was misuse engineered by US Americans, to make themselves the "main" America. But for many reasons I think the context is very different in Europe, especially since the obvious grab by EU institutions hasn't really worked among Europeans, even EU Europeans.
EDIT: Further to that "Pan-American" is well understood to not just mean the USA, so "Pan-European" cannot possibly mean the EU only except by very poor wording choices or a very political agenda.
I don't "live in that context", I'm aware of it can can use it when appropriate.