Yes, $U(1)_{PQ}$ has standard model anomalies. The difference is that
in QCD
$$
{\cal L}=f_\pi^2(\partial_\mu \eta')^2 - V(\eta'-\theta)
$$
where $f_\pi$ and $V^{1/4}$ are both $O(\Lambda_{QCD})$, so there is no sense in which the anomaly is a small effect. In particular, you cannot argue that there is a spontaneously broken $U(1)$ which is weakly broken by the anomaly. Indeed, the mass of the $\eta'$ is $m\sim 1$ GeV, comparable to the mass of the proton.
[There are exceptions to this statement, even in QCD. You can consider QCD at very high baryon density. Then $f_\pi\sim \mu \gg \Lambda_{QCD}$ and $V\sim \Lambda_{QCD}^4 (\Lambda_{QCD}/\mu)^c$ where $c$ is a positive power. Then the $\eta'$ is light and we can talk about spontaneous $U(1)$ breaking, weakly broken by the anomaly. A similar situation exists in QCD at large N.]
In axion extensions of the SM we have
$$
{\cal L}=f_a^2(\partial_\mu a)^2 - V_a(a-\theta)
$$
where $V_a$ comes from SM anomalies (mostly QCD), so $V\sim \Lambda_{QCD}^4$. But $f_a$ is a free parameter, so we can make it as large as we want (phenomenologically it has to be much bigger than $\Lambda_{QCD}$). Then $m_a^2=\chi/f_a^2$ where $\chi\sim\Lambda_{QCD}^4$ and $m_a\ll \Lambda_{QCD}$,
and we have an approximate $U(1)_{PQ}$.
Final Remark: What do we mean by approximate spontaneous symmetry breaking? The standard example is the $SU(2)_L\times SU(2)_R$ symmetry of QCD. This symmetry is spontaneously brokken by the quark condensate, $\langle \bar{\psi}_L\psi_R\rangle \simeq -\Lambda^3_{QCD}$, and explicitly broken by the quark masses $m_q$. If $m_q=0$, then chiral symmetry is exact amd sponatenous breaking leads to massless Goldstone bosons. If $m_q\neq 0$ then Goldstone bosons acquire a mass $m_{GB}^2\sim B/f_\pi^2$ with $B=m_q\langle\bar\psi_L\psi_R\rangle$. As long as $m_{GB}\ll \Lambda_{QCD}$ we can view the effect of the quark mass as a small perturbation, and talk about the spontaneous breaking of an approximate symmetry. If $m_{GB}\sim \Lambda_{QCD}$ there is no sense in which an approximate symmetry is broken.
