Do math PhDs - post-doctoral researchers have an advisor? Typically, well-prepared undergrad math students have summer REU advisors, masters and PhD students have thesis advisors.  

What do new math PhDs - postdoctoral researchers have at the university they land at?  Do they also have a research advisor, or are they pretty much on their own, while they teach classes, publish papers and apply to tenure-track positions, before their post-doc contract expires?
This varies greatly, depending on the type of postdoctoral position.  Some postdocs are funded by a specific grant.  Then they have a supervisor, namely the PI on the grant.  How formal the supervision is can vary, but someone is at least nominally in charge of the postdoc, and there's an expectation that the postdoc will do something relevant to the grant.

Other postdocs are hired by the department or a mathematics institute, rather than a particular PI (this is especially common in pure mathematics).  Often the postdoc is assigned a mentor, but what's involved in the mentoring relationship can vary.  It could consist of collaborative research, serious professional guidance, or just chatting occasionally.