Could a 'polymer' be formed by hydrogen bonding?
Since a polymer is a

&quot;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.&quot; (IUPAC Gold Book)

joining two monomers into a dimer is just a specific wording for the more general term polymer like it is for trimer, tetramer, pentamer, hexamer, etc.  While the term polymers often is used about plastics, cellulose, starch, or chitin equally are polymers, too.  Note, this definition does constraint the binding of the monomers to covalent bonds.
The depictions shown as examples by you are valid representations when describing, e.g., the hydrogen-bond pattern in organic crystals.  Here, hydrogen bond donors and hydrogen bond acceptors may contribute significantly to the overall stabilization of the supramolecule an organic crystal may be considered.  Magaret C. Etter and Joel Bernstein popularized  formalisms to identify and systematically describe these pattern.
The one described by you matches for example the situation for $p$-terephthalic acid yielding (in principle) infinite bands hold by the motif $R^2_2(8)$ (a ring of eight atoms in total, comprising 2 donor, and 2 acceptors):

(1990AccChemRes120, includes a guide how to define these graph sets)
This approach equally may be extended to describe the interactions if the monomers are chemically different:

($R^2_4(8)$ motif about the interaction of phosphine / arsine oxides and  nitroanilines in co-crystals, reviewed in 1995ACIE1555)