When do we consider states under a $U(1)$ transformations to be physically different? Consider the Goldstone model of a complex scalar field $\Phi$. It has $U(1)$ global symmetry, so if
we apply the transformation $\Phi \to e^{i\alpha} \Phi$ the Lagrangian is left invariant. 
Furthermore, we have an infinite set of possible vacua all with the same non-zero vacuum expectation value.
But the vacuum changes under a $U(1)$ transformation, so $U(1)$ symmetry is spontaneously broken.


*

*In this case we assume that for each value $\alpha$, 
$e^{i\alpha} |0 \rangle$ corresponds to a different state,
right?

*But since we can't measure a phase, wouldn't it be more natural to
consider them the same states? On the other hand, if I think of the
real and imaginary parts as two independent fields, I would say that
they shouldn't be the same states.


Let's now couple $\Phi$ to a gauge field, such that the Lagrangian is invariant under local $U(1)$ transformations.
Then we do consider  $e^{i\alpha} | \Psi \rangle$ to be all the same states, right?  But then, doesn't this mean that the Higgs vacuum is unique?
 A: You are wrong from this assumption:

We assume that for each value $\alpha$, 
     $e^{i\alpha}\Phi | 0 \rangle$ corresponds to a different state.

The wave function of the symmetry broken vacuum (the Higgs vacuum) is not $e^{i\alpha}\Phi | 0 \rangle$ as you assumed, but a coherent state $|v\rangle$ (a boson condensate), obtained by applying the shift operator $D(v)$ to the symmetric vacuum state $|0\rangle$:
$$|v\rangle=D(v)|0\rangle=e^{v\Phi^\dagger-v^*\Phi}|0\rangle,$$
where  $v=|v| e^{\text{i}\alpha}$ is the vacuum expectation value (VEV) of the boson operator $\Phi=\int\text{d}^dx\Phi(x)$. Note that in the Higgs vacuum state $|v\rangle$, the operators $\Phi$ and $\Phi^\dagger$ are raised to the exponent and the phase factor $e^{\text{i}\alpha}$ is not exposed as an overall factor of the wave function. One can indeed show that $\forall x:\langle v|\Phi(x)|v\rangle=v$ using the boson commutation relation $[\Phi(x),\Phi(x')^\dagger]=\delta(x-x')$ and the definition of the symmetric vacuum state $\Phi(x)|0\rangle=0$ (see this Wikipedia page for more about coherent states). The Higgs vacuum state transforms nontrivially under the U(1) symmetry action: $|v\rangle\to|ve^{\text{i}\theta}\rangle$. The new vacuum is orthogonal to the old one $\langle v|v e^{\text{i}\theta}\rangle=\delta(\theta)$ (in the thermodynamic limit). So the two states are indeed physically different under the U(1) transformation.
