Why in the Ridge regression, the coefficients cannot be 0? In the second answer (https://stats.stackexchange.com/a/368426/287815) to the question (Why will ridge regression not shrink some coefficients to zero like lasso?), the OP found out that,
$β = /(^2+λ)$
My question:
Can't the $\beta$ be 0 if one of x or y is a zero vector?
 A: What the referred answers say is that in ridge regression the regularization does not shrink the parameters to exact zeros. It doesn't say that the parameters cannot be zeros, just that it's not what the regularization does. Your example with zero vectors is a pathological case and even without regularization, the parameter would be zero, so it has nothing to do with regularization.
A: 
what is the difference between "shrinking to exact 0" and "parameters can be 0"

If the OLS solution is non-zero (which means that $y$ is non-zero) then the ridge regression regularisation will not be able to shrink the parameters to exact 0.
As you found out the parameters of ridge regression can be zero when $y=0$ ($x=0$ makes no sense*). But that is only the case when $y=0$ and in that case the OLS solution is also zero (so it is not non-zero and there is nothi g to shrink). For this case it is not due to shrinking that the ridge regression parameters are zero, but due to the OLS solution being already zero.
BTW when you observe a continuous variable then the case of a zero $y$ vector and a zero coefficient will have zero probability.

* The vector $x$ won't be zero. Because that would be a useless model $y = \beta \cdot 0$.
