There are actually two ways to answer this question: by lawyering or by trying to capture what a polymer is. I'll try both.
Answer by lawyer
The IUPAC defines the following terms as follows:
- monomer: a molecule which can undergo polymerisation thereby contributing constitutional units to the essential structure of a macromolecule
- macromolecule: A 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.
- molecule: An electrically neutral entity consisting of more than one atom ($n > 1$). Rigorously, a molecule, in which $n > 1$ must correspond to a depression on the potential energy surface that is deep enough to confine at least one vibrational state.
Using the data from this paper and a harmonic oscillator approximation, I estimate that the zero-point energy of a harmonic oscillator with two waters and a hydrogen bond is just over 45 kJ/mol.
Since the energy contributed by hydrogen bonding in liquid water is estimated to be between 2kJ/mol and 6kJ/mol, I conclude that the hydrogen bond of water cannot confine a vibrational state. This means liquid water does not qualify as a single molecule.
Since a long chain of waters connected by hydrogen bonds does not qualify as a molecule, it cannot be a polymer, which means that water cannot be a monomer (at least, not in this sense).
A better answer
The real problem here isn't the technical stuff I wrote above (which, by the way, probably has a few holes large enough to sneak trucks through). It's that words have certain meanings, and we can't capture everything in a simple definition. For the same reason that you'd probably feel a bit weird calling "yellow" a "reddish-green", even though it is literally the result of mixing red and green light, it feels strange to call water a polymer.
Here are some common ideas that spring to mind when I see the word "polymer":
- Made of organic monomers bonded together covalently
- Structure of covalent bonds (e.g. branching) affects physical properties
- Intermolecular forces affects physical properties
These are not strict, but they explain why quartz is not referred to as a polymer, even though it's made of covalently-bonded $\ce{SiO2}$ units: everything is covalently bonded and there aren't any intermolecular interactions to speak of! For another example, depending on the exact context, proteins are sometimes rarely thought of as polymers, even though they're made of repeating amino acids, because their structure is fixed (must be linear chain) and their intermolecular forces don't contribute much.
So the more sensible answer to this question that doesn't rely on lawyering is that liqid water just doesn't have the properties that most people think of when they hear the word "polymer" and so it isn't usually referred to as a polymer.
Supplemental Materials
The code I used to compute the harmonic oscillator energies
from math import pi,sqrt
# Compute the harmonic oscillator energy of two water molecules
# using a two-body QHO approximation
kg_per_amu = 1.66054e-27
watermass = 18.0 * kg_per_amu # Mass of water: close enough for now
reducedmass = watermass**2 / (2 * watermass)
hbar = 1.054571800e-34 # J-s
k = 845.4 # From https://www.princeton.edu/~fhs/paper79/paper79.pdf, N-m
E_zeropt = hbar * pi * sqrt(k / reducedmass) # for n = 0
E_zeropt_kjm = E_zeropt * 6.022e23 / 1000 # Scale from J/molecule to kJ/mol
print(E_zeropt_kjm)
