We know that gravitational waves stretch space-time so light travelling through that space should be stretched as well. So my question is how do we know anything is stretching if the effect is experienced by every object in the universe.
I understand your question and wrestled with it myself for some time.
The reason that the Michelson-type interferometer setup works is that the travel time of the photons in each arm is much shorter than the time period of the gravitational wave. Thus the photons are not really stretched by the gravitational wave at all - they are simply emitted into an arm that has changed its apparent length. The effect is different in the two perpendicular arms (one is stretched whilst the other is contracted) and this produces a phase difference in the signal when the beams in the two arms are recombined. The phase difference is modulated with the frequency of the gravitational wave (which is much lower than the frequency of the laser light).
For example, the LIGO arms are effectively 1120 km long, because the light is bounced backwards and forwards 280 times. Light spends 3.7 micro-seconds in the apparatus. The gravitational waves that are being detected have frequencies below a kHz, so the arm length only changes significantly on times longer than a milli-second. Thus the photons "see" arms of fixed length.
Geometry of gravitational waves is such, that it stretches and shrinks spacetime differently in perpendicular directions. So the detector notices the difference in flight time between two photons going along different axes of the detector, as the velocity of light is constant. "Stretching" of a photon would mean, that it changes color (wavelength) but not flight time.
If you're interested in the math have a look at The Geometry of Gravitational Wave Detection by John T Wheelan.