The big difference is that most places in Europe don't have a tradition of assistant professorship, or, more generally, tenure-track faculty. Here's an example of the "typical" career progression of an academic in Austria (durations can vary by a lot, I just give examples here for illustration purposes):
- Phd Student (4 to 5 years finished with Doctorate)
- Postdoc (6 to 10+ years finished with habilitation / venia docendi)
- "Privatdozent" (undefined length, non-tenured position, somewhat comparable with a non-tenured associate professor)
- Full Professor / Chaired Professor (tenured position).
Note that there is no assistant professorship, and (2) that no tenured position before Full Professor exists.
Clearly, in this system there are postdocs with many years of experience in their field. They often have developed their own small group, work very loosely with a faculty mentor or completely independently, have their own funding and supervise PhD students that are "quasi" theirs (postdocs are not formally allowed to supervise PhD students before habilitation, but it is common practice that the mentor of the postoc formally supervises the postdoc's staff without involving her/himself much in the actual process). These postdocs are often considered assistant professors, even going so far as to call themselves assistant professor on their web pages and business cards despite not actually being professors de jure.
The general takeaway is that being a postdoc in Europe may have higher status than in the US, but it is not true for every postdoc. As postdoc status can last a very long time in Europe, one needs to look very closely at the person himself to see how senior she/he actually is.