  The Standard Model (SM) without the Higgs mass term is scale invariant.
Gildener and Weinberg generalized the scale invariant standard model (SISM) by
including the multiplication of scalars in quartic forms. They pointed out that
along the flat direction only one scalar -- called the scalon -- is classically
massless and all other scalars are massive. Here we choose a SISM with one
scalon and one heavy scalar and extend that further respecting the scale
invariance by a vector-like lepton (VLL). By an appropriate choice of the flat
direction, the heavy scalar enjoys the $\mathbb{Z}_2$ symmetry and is assumed
as DM particle. The scalon connects the visible and dark sector via the
Higgs-portal and by interacting with both the muon lepton and the VLL. The VLL
is charged under $U(1)_Y$ and interacts with $\gamma/Z$ bosons. We show that
the model correctly accounts for the observed dark matter (DM) relic abundance
in the universe, while naturally evading the current and future bounds from
direct detection (DD) experiments. Moreover, the model is capable to explain
the $(g-2)_\mu$ anomaly observed in Fermilab. We also show a feature in SISM
scenarios which is not present in other Higgs-portal models; despite having the
Higgs-portal term $|H|^2 s^2$ ($s$ being the scalon) in SISM, the effective
potential after the electroweak symmetry breaking lacks an important expected
vertex $h s^2$. This property immediately forbids the tree-level invisible
Higgs decay $h\to ss$ and the one-loop Higgs decay $h\to \mu^+ \mu^-$.
