Beam-splitters are not designed for any specific phase shift, nor does anyone actually model different kinds of beam-splitters (half-slivered, cube, pellicle etc.) differently in a high level description of an optical setup. The choice of phase is entirely a mathematical convention.
The simplest way to understand this is by trying to come up with an experiment where you could measure the phase, and thereby distinguish beam-splitters with different phase shifts. First of all, we need to be clear about what phase we're talking about. For a general balanced beam-splitter we have:
$$
E_3 = \frac{1}{\sqrt{2}}(E_1 + e^{i\theta_2}E_2)\\
E_4 = \frac{1}{\sqrt{2}}(e^{i\theta_1}E_1 + E_2)
$$
As shown in the answer by G.C. to the question you linked, the phases $\theta_1$ and $\theta_2$ have to add to $\pi$ in order for the beam-splitter to conserve energy: $\theta_1+\theta_2=\pi$, so simply determining one of them is sufficient. Let's say we want to determine $\theta_2$, which is the relative phase between the transmitted and reflected light in (the output) port '3'. In an experiment, what we want to do is therefore to send light into the two input ports '1' and '2', and observe the interference of this light in port '3', as this would tell us the relative phase between the reflected and transmitted beams.
However, there's a problem with this approach, because the two input fields $E_1$ and $E_2$ have a relative phase as they hit the beam-splitter, so the output in mode '3' is actually:
$$
E_3 = \frac{1}{\sqrt{2}} (e^{i\phi_1}E_1 + e^{i(\phi_2+\theta_2)}E_2),
$$
where $\phi_1, \phi_2$ are the phases of the input fields on the beam-splitter.
To perform the experiment we therefore first need to measure and fix these phases. But how do we measure optical phases? By interfering the fields! In other words, taking a beam-splitter, combining the two beams and observing the output; when one port is dark the fields are in phase, and when the other port is dark the fields are out of phase. In this type of experiment we can't distinguish between the contribution from the input phase and the contribution from the beam-splitter. Therefore, we simply fix the beam-splitter phase by picking a convention, and then define the optical phase relative to this.
To some extent this is a practical limitation, because we at present do not have measurement devices capable of directly detecting the optical phase, and even if we did the short wavelength of light means that it's infeasible to compare the phases of optical fields at two separate points. When dealing with lower frequency fields (like radiowave frequencies) it is possible to directly measure and compare phases without interfering them.
Note that there are situations in optics where the actual reflection phase matters, such as in anti-reflection coatings using thin films. In these cases the phase can be measured assuming you know the thickness of your film to sufficient precision.
