Is benzene a polymer of ethyne?
The scientist who coined the term polymer, Jöns Jacob Berzelius, used the word to refer to different substances which had the same empirical formula. In this sense, benzene is a polymer of ethyne (acetylene), because benzene $\ce{C6H6}$ and ethyne $\ce{C2H2}$ have the same empirical formula, namely $\ce{CH}$. In other words, all substances with molecular formula $\ce{(CH)_x}$ were polymers of the smallest unit in the family (in this case, $\ce{C2H2}$, because $\ce{CH}$ molecules are only known as extremely fleeting species on Earth, though they are abundant in space).
However, that was back in the 1830s, when we had no clue about atomic and molecular structure (in 1808, one of the eminent chemists of the time, John Dalton, defended $\ce{HO}$ as the formula for water). The meaning of the term has evolved over time to accommodate our increase in knowledge. The most current IUPAC recommendations on polymer nomenclature (Compendium of Polymer Terminology and Nomenclature - IUPAC Recommendations 2008) states the definition of polymer as:

A polymer is a substance composed of macromolecules.

And, fortunately, they also define a macromolecule as:

Molecule of high relative molecular mass, the structure of which essentially comprises the multiple  repetition  of  units  derived,  actually  or  conceptually,  from  molecules  of  low  relative molecular mass.


Note  1:  In  many  cases,  especially  for  synthetic  polymers,  a  molecule  can  be  regarded  as  having  a  high  relative  molecular  mass  if  the  addition  or  removal  of  one  or  a  few  of  the  units has a negligible effect on the molecular properties. This statement fails in the case of certain properties of macromolecules which may be critically dependent on fine details of the molecular structure, e.g., the enzymatic properties of polypeptides.


Note  2:  If  a  part  or  the  whole  of  the  molecule  has  a  high  relative  molecular  mass  and  essentially  comprises  the  multiple  repetition  of  units  derived,  actually  or  conceptually,  from   molecules   of   low   relative   molecular   mass,   it   may   be   described   as   either   macromolecular or polymeric, or by polymer used adjectivally.


Note  3:  In  most  cases,  the  polymer  can  actually  be  made  by  direct  polymerization of  its  parent  monomer  but  in  other  cases,  e.g.,  poly(vinyl  alcohol),  the  description  ‘conceptual’  denotes that an indirect route is used because the nominal monomer does not exist.

As you can see, there is quite a bit of detailing to try to keep things consistent. Observe that the definition speaks of high relative molecular mass, and this point is expanded upon in Note 1. With these considerations, benzene is not a polymer - it is certainly a very small molecule, and adding or removing a portion of it (presumably cyclooctatetraene $\ce{C8H8}$ and cyclobutadiene $\ce{C4H4}$, respectively) would result in a substance with drastically different properties.
In some sense, Berzelius wasn't really wrong to call benzene a polymer, because for all he knew back then, benzene could well have the formula $\ce{C20000H20000}$ (actually, it was not widely believed that &quot;large&quot; molecules could exist at all, with some resistance to the idea existing as late as the 1920s-1930s). He did the best with what knowledge was available to him. But now we can do better, and it just is more convenient and self-consistent to not label benzene as a polymer.
This isn't to say that, however, that benzene and ethyne are completely chemically unrelated. There is in fact a close relationship, in that ethyne can efficiently undergo a [2+2+2] cycloaddition reaction to form benzene. This is not exploited industrially, however, as there are much cheaper (and safer) ways of making it.