>there's no a European common structure.
There are European corporate entities nowadays. See SE[0], SCE[1] and SPE[2]. The SPE is what you'd call an LLC, GmbH, SARL etc.
[0] https://en.wikipedia.org/wiki/Societas_Europaea
[1] https://en.wikipedia.org/wiki/Societas_cooperativa_Europaea
But another benefit from LLCs in the USA might be the lighter paperwork and simpler taxes filing.
Which we also have (in France at least, as entrepreneur individuel), but both don't intersect.