Electric field vector in visualization of polarization of EM wave When we were taught polarization at the high school level, we were told that during polarization, we should consider the EM wave being axially or planarly filtered (e.g. by a polarizing sheet with slits) in terms of its electric field vector only (i.e., if the EM wave was a wave on a string, the electric field vector's oscillations would be the wave on the string), ignoring the effect on the magnetic field vector, for some reason which our teacher didn't know. Why is this done?
I once read somewhere that only the electric field vector of visible light stimulates our eyes. Does that have anything to do with this consideration?
As a side note, our textbook had an illogical explanation for this, they equated the energy density of the electric and magnetic fields, and from there deduced that the magnetic field is much weaker than the electric field. Surely this is like saying that 1s more than 1m, and each represents a different quantity, so they should be incomparable.
 A: Let's start with some background information:
A typical human eye will respond to wavelengths from about 390 to 700 nm. In terms of frequency, this corresponds to a band in the vicinity of 430–770 THz. This means our eyes will respond to or more accurately absorb photons in this wavelength range  and frequency band. A photon is an ElectroMagnetic wave. It is not one or the other. When the photon and its energy quanta is absorbed it means the photon and its EM field(s) and $E=h \nu$. Consider The Electrcic and Magnetic fields as one, an EM field.   
Now let's address your questions -
When we were taught polarization at the high school level, we were told that during polarization, we should consider the EM wave being axially or planarly filtered (e.g. by a polarizing sheet with slits) in terms of its electric field vector only (i.e., if the EM wave was a wave on a string, the electric field vector's oscillations would be the wave on the string), ignoring the effect on the magnetic field vector, for some reason which our teacher didn't know. Why is this done?
Really it is done arbitrarily since the magnetic field oscillates at 90 degrees from the electric field, so either could be chosen. It has become the standard to look at the the electric field's vector for reference to polarization. {But the magnetic field vector could have been used as well} 
So in phsyics when we look at the axial ratio to describe electromagnetic radiation with elliptical, or circular, polarization. The axial ratio is the ratio of the magnitudes of the major and minor axis defined by the electric field vector. Looking at one vector is much simpler than looking at both.
Next -
I once read somewhere that only the electric field vector of visible light stimulates our eyes. Does that have anything to do with this consideration?
To be accurate we need to say that it is the absorbed photon, of $E=h \nu$ and specific wavelength, and that is stimulating our cone and rod cells in our eyes, not an electric field vector. Again the EM wave, photon as one system. The EM energy of the photon is being converted into electrochemical energy and the nerve cells are sending a message to the brain.
Now let's sum up what we are saying. The entire photon's system, its physical, electric and magnetic fields are absorbed during visualization not just one or the other and 
the electric field vector is arbitrarily used to describe the axial ratio for circular or elliptical polarized light it is also used for a point of reference for planar polarized light.
