A search for dark matter particles produced in association with a Higgs boson decaying to a bottom quark-antiquark pair in proton-proton collisions at
s
=
13
TeV
is presented. The data, collected with the CMS detector at the LHC, correspond to an integrated luminosity of
101
fb
−
1
. The analysis is performed in exclusive categories targeting both Lorentz-boosted (merged) and resolved
b
jet pair topologies, covering a wide range of Higgs boson transverse momentum. A statistical combination is made with a previous search using data collected in 2016 and corresponding to an integrated luminosity of
35.9
fb
−
1
. The observed data agree with the standard model background predictions. Constraints are placed on models predicting new particles or interactions, such as those in the simplified frameworks of baryonic-
Z
′
and
2
HDM
+
a
, where the latter is a type-II two-Higgs-doublet model featuring a heavy pseudoscalar with an additional light pseudoscalar. Upper limits at 95% confidence level are set on the production cross section for these models. For the baryonic-
Z
′
model,
Z
′
boson masses below 2.25 TeV are excluded for a dark matter particle candidate mass of 1 GeV. In the
2
HDM
+
a
model, heavy pseudoscalar masses between 850 and 1300 GeV are excluded for a light pseudoscalar mass of 350 GeV.