Do Postdocs have any seniority ranking or progress? I am seeing many people doing postdocs for years, or even nearly a decade, well into their late thirties. Do they usually have progressive salary and responsibility, as one could expect in a job?  Is your fifth or sixth year as a postdoc essentially the same as your first month after PhD., but with some extra publications? 

For clarification, this is what I mean by progress. 

In your later years, do you get to build and lead a small team of junior researchers who you hire from your grant, whom you can give some instruction, or you divide the tasks of a more ambitious project and you take some supervisory role in addition to your own hands-on research? I do not expect it to be completely managerial because I know postdocs love the actual research…but at least some mark of seniority and experience?
Of all the kinds of academic positions, the postdoc is probably the most variable.  The common ground across (almost) all postdoctoral positions is: you should have a doctoral degree when you start the position, and the position should not be permanent or tenure-track.  That's really about it.  (Added: I agree with @StrongBad that most postdoctoral positions have an expectation of "advanced training" of one kind or another.)   

It is rare for a single postdoc position to last more than 4-5 years.  In my understanding, the ones that do are soft-money positions and thus are more explicitly doing specified projects for the PI / grantholder.  I am not aware of any kind of formal seniority across multiple postdocs: generally the reason that someone does a second postdoc is because they want a permanent academic job and haven't gotten one yet.  They may also want additional specific training / the chance to work with other researchers, but this really is "additional" rather than on top of.  

In terms of salary: in my experience [mathematics, US], postdoc salaries are rather flat across the period of employment.  (I previously had more detailed information, but I am less confident that it is representative even among math postdocs in the US.)  In my department we have one kind of postdoc that pays about $10K per year more than another postdoc is funded by an NSF grant, and the salaries for the latter postdoc are set by the NSF, so will be the same unless / until we get a new grant (or an updated version of the same).  

When it comes to skilled labor, wherever you work for a year or more, you gain knowledge, experience and seniority in some informal sense.  One might say that a postdoc is a process of laundering a student into a faculty member, so late-career postdocs resemble faculty members more than early career postdocs.  Most postdocs in mathematics find their way from a combination of completing / continuing their thesis work and starting new projects with their new supervisor to developing a distinctive and independent research program.  Although the particulars differ across fields, I think that something like this is the goal of most postdocs: if a postdoc only does their supervisor / PI's work then it will be difficult for them to portray themselves as a valuable faculty hire.

For math postdocs, the only formal seniority comes from teaching obligations: namely most math postdocs start out teaching freshman / sophomore level courses.  If they are competent at their teaching and interested in teaching other courses (not all postdocs are), many postdocs move on to teach a junior/senior level course for majors in their second year and some postdocs teach a graduate course in their last year.  (In my fifth semester as a postdoc I was able to teach a topics course in the arithmetic of Shimura varieties.  I have since taught about a dozen graduate courses, but none as advanced as that.)


  In your later years, do you get to build a small team of junior researchers who you hire from your grant whom you can give some instruction,, 


Some postdocs are not associated to a grant in any way; they are rather temporary faculty funded by their department.  Some postdocs work under their PI's grant, in which case the answer to the question is up to the PI (and the grant proposal).  Some postdocs receive grant funding of their own, in which 
case some kind of money for junior personnel may well be included.  Also some postdocs work with graduate students, in a variety of different ways.  My current postdoc is for instance a member of the dissertation committee of one of my students.


  or you divide the tasks of a more ambitious project and you take some supervisory role in addition to your own hands-on research? 


That could happen.  Some large research groups employ postdocs in a kind of managerial role.


  I do not expect it to be completely administrative or managerial because I know postdocs love the actual research…but at least some mark of seniority and experience? 


Keep in mind that any seniority you acquire as a postdoc is ephemeral.  Spending too much time doing administrative / managerial work as a postdoc sounds very risky if you are aiming for a permanent faculty position.  Some departments somewhere must be hiring some faculty largely for their administrative skills...but I confess I have never seen it with my own eyes.