  The tension among measurements of $V_{us}$ from different channels, the
so-called Cabibbo Angle Anomaly, can be interpreted as a signal of lepton
flavor universality (LFU) violation in the $W$ boson couplings. We investigate
this issue in the framework of effective field theory, keeping the gauge
structure of the Standard Model (SM) unchanged. We introduce gauge-invariant
dimension-6 effective operators that couple the Higgs doublet to leptons,
thereby giving non-universal tree-level contributions to the couplings of
electroweak gauge bosons. Due to the $SU(2)_L$ gauge symmetry, a tension arises
between the $V_{us}$ measurements that are affected by new $W$ couplings, and
the electroweak precision measurements, which are also affected by the new $Z$
couplings. We show that this tension can be alleviated by allowing additional
sources of gauge-invariant couplings of $Z$ boson to left- or right-handed
leptons, and find the optimal regions indicated by the current data in the
Wilson-coefficient space. We illustrate our model-independent results with the
examples of minimal extensions of the SM involving the vector-like lepton (VLL)
models. We point out that dimension-6 operators coupling the Higgs doublet to
leptons can affect the rate of $h \to \tau \tau$ decay significantly in
general, however this effect is restricted to less than a per cent level for
the minimal VLL models.
