  We show that 1D electron states confined at twin-grain-boundaries in
MoSe$_{2}$ can be modeled by a three-orbital tight binding model including a
minimum set of phenomenological hopping terms. The confined states are robust
to the details of the defect hopping model, which agrees with their
experimental ubiquity. Despite a valley Chern number which is finite and
opposite on both sides of the defect, there is no topological protection of the
confined states. This turns out to be an essential feature to have only one
confined electronic band, in agreement with experiments, instead of two, as the
bulk-edge correspondence would imply. Modeling the confined state as a 1D
interacting electronic system allows us to unveil a mobile quantum impurity
type behavior at energy scales beyond the Tomonaga-Luttinger liquid with an
interaction range which extends up to the lattice spacing, in excellent
agreement with ARPES measurements.
