How do we know that x amount of strain equals x amount of pressure?
Strain gauges have seen extensive testing for decades. When used to measure strain in a system for which the strain is known, there is good agreement.
In a barrel (ie. a steel tube), the equations to predict strain from pressure (and vice verse) were developed long ago. There are some errors introduced by the forces on the barrel near the receiver and by recoil. Both are
considered to be small.
The trick for the home user is to apply the gauge well and to measure the dimensions of the metal on which it is applied. Errors in either mean the calculated pressure will be off.
Another complication is the brass. It acts to reduce the area over which the pressure acts, and it is also applies a resisting force to contain some of the pressure. You must, then, make careful measurements of the brass thickness under the applied gauge. Lastly, you use an average value for the strength of the brass to calculate its resisting effect.
There are several "ifs" here, but Ken Oehler has written that in practice it all works out pretty well. (If I recall, he said one gets decent results even if you use a set psi offset to compensate for the brass.)
I have read that strain gauge systems are calibrated with ammo that produces known pressures
No ammo produces a known pressure in every chamber. You'll see claims on the web that "SAAMI reference" ammo exists for this purpose, but it is not so. Reference ammo is extremely consistent ammo that has been tested in several pressure barrels known to be correctly cut. Such ammo is used to qualify a newly made pressure testing barrel. If the new barrel is correctly cut and if the sensors are right, then the reference ammo should produce the same psi and fps as in the existing test barrels. It's simply a check on the new system; it is not proof of accuracy.
At home, it's safest to use a variety of factory ammo to set an upper limit for the measured strain. With this, one doesn't care about the calculated pressure. If your gun is safe with factory ammo, handloads made to the same strain reading should be as well.
One must also use a chronograph as a check that the indicated pressures aren't too good to be true.