  The measurements of $R_{K^{(*)}}=\mathcal B(B\to
K^{(*)}\mu^{+}\mu^{-})/\mathcal B(B\to K^{(*)}e^{+}e^{-})$ in recent years have
hinted lepton flavor non-universality and thus drawn widespread attentions. If
these anomalies are induced by new physics (NP), deviations from the SM
predictions may also be found in other channels via the same process at the
quark level. In this work, we study in $B\to K_{1}(1270,1400)\mu^{+}\mu^{-}$
decays the effects of two popular classes of NP models which can address the
$b\to s$ anomalies, i.e. the leptoquark models and the $Z^{\prime}$ models. By
assuming that NP only affects the $b\to s\mu^+\mu^-$ transition, we find that
the unpolarized and polarized lepton flavor universality (LFU) ratios
$R_{K_1^{(L,T)}(1270)}$ are useful to distinguish among the NP models
(scenarios) and the SM because they are sensitive to the NP effects and
insensitive to the mixing angle $\theta_{K_1}$, while the
$R_{K_1^{(L,T)}(1400)}$ are sensitive to both NP and $\theta_{K_1}$. Another
ratio $R_\mu(K_1)=\mathcal B(B\to K_1(1400)\mu^+\mu^-)/\mathcal B(B\to
K_1(1270)\mu^+\mu^-)$ is shown to depend weekly on the effects from the NP
models (scenarios) under consideration, and thus can be used to determine the
$\theta_{K_1}$ and complement the $R_{K_1^{(L,T)}}$ in the probe for NP.
