Must we test whether e.g. $A=B$ and $A=C$ implies $B=C$ by experiment? Chaper 10, conservation of momentum in "The Feynman Lectures on Physics" in the chapter entitled, the authors write that

Suppose we know from the foregoing experiment that two pieces of
  matter, $A$ and $B$ (of copper and aluminum), have equal masses, and we
  compare a third body, say a piece of gold, with the copper in the same
  manner as above, making sure that its mass is equal to the mass of the
  copper. If we now make the experiment between the aluminum and the
  gold, there is nothing in logic that says these masses must be equal;
  however, the experiment shows that they actually are. So now, by
  experiment, we have found a new law. A statement of this law might be:
  If two masses are each equal to a third mass (as determined by equal
  velocities in this experiment), then they are equal to each other.
  (This statement does not follow at all from a similar statement used
  as a postulate regarding mathematical quantities.) 

I assume that by "a similar statement used as a postulate regarding mathematical quantities" the authors refer to the transitive axiom of algebra, that is, if $A=B$ and $A=C$, then $B=C$.
It seems to me that, using Feynman's definition of mass and the transitive axiom of algebra, one must conclude that it is the case that $B=C$ even without making an experiment. Why do Feynman et al claim that 

there is nothing in logic that says these masses must be equal; however, the experiment shows that they actually are

 A: Who can with certainty fathom the mind of a Feynman? :-)
However, it sounds like he is saying that there is no reason to decide with certainty that Gold, Copper and Aluminum do not have properties which affect how they behave when accelerated by a force.
An example that has some correspondence is carrying out the same experiment in air.
Mount 1 kg spheres of each metal on massless, frictionless, area less "carts" (made of unobtainium (or nonexisteum) ) such that they can be slid across a surface in a manner which is independent of the carts' properties. Now place any two spheres in contact on their carts and apply an explosion based force as before. The initial velocities will be the same as before and equal but some mysterious new force will apply that causes the spheres to act differently. Gold-Al and Gold-Cu will have different results for the non Al material. A few nano-seconds thought will allow us to see that in both cases the Al sphere behaves the same way and that there is some property of each material that affects how they behave once proceeding at a given velocity. We will have discovered air resistance and drag and the different densities will cause the different masses to have non-equal frontal areas so they will have different deceleration forces from drag. 
While this is so obvious that it seems trivial, there may have been other properties of Au.Al/Cu that affected how they interacted in the initial experiment. Until we perform the experiment we cannot know what curve ball nature has or hasn't got in store.
This leads to his extremely valuable observation
   "From this example we can see how quickly we start to infer things if we are careless."

If we were unaware of air drag then expecting equal behaviour for all 3 materials after the initial explosion would leave us puzzled. That we are not puzzled in the free space case is because our expectation happened to ,match reality. But, we can never be sure that this is the case.

Real (out of this) world example:
For some while it seemed that there were 1/3 as many Neutrinos coming from the sun as calculations predicted. In due course it was posited and then experimentally "proven" that the Neutrinos 'change colour' as they amble on their way at (or just above or just below :-) ) light speed, and that only the 1/ which are the 'right' colour as they pass through our detectors get detected.
Solar neutrino problem
From Wikipedia - Solar Neutrinio Problem


*

*Measurements of solar neutrino types were not consistent with models of the Sun's interior.

*Former Standard Model: Neutrinos should have been massless according to the then-accepted theory; this means that the type of neutrino would be fixed when it was produced. The Sun should emit only electron neutrinos as they are produced by H–H fusion.
Observation

*Only one third to one half of predicted number of electron neutrinos were detected; neutrino oscillation explains the difference but requires neutrinos to have mass.

*Resolution: Neutrinos have mass and so can change type.
From Wikipedia - Neutrino oscillation


*

*Neutrino oscillation is a quantum mechanical phenomenon whereby a neutrino created with a specific lepton flavor (electron, muon or tau) can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates through space.

*First predicted by Bruno Pontecorvo in 1957, neutrino oscillation has since been observed by a multitude of experiments in several different contexts; it also turned out to be the resolution to the long-standing solar neutrino problem.) 
A: I think the content of his statement is that "the assignment of mass to an object as defined using the conservation of momentum experiment is a transitive property". Not all relations are transitive properties (for instance, "Wolves eat Deer, Deer eat grass, but wolves don't eat grass"). For instance, if we had defined "the mass of an object" as a weight $mg$, the transitive property might not be satisfied unless the experiment was performed at the same location on Earth.
So in my mind, the story from Feynman is a statement about how mass is well-defined using the conservation of momentum. When we say "the mass of the car is 500 kg", we hope that can mean everything that the statement "$m_{car}=500$ kg" does, and when we perform the experiment we have some evidence that it is at least transitive, as it should be.
A: We know how real numbers (in the mathematical sense) behave. There is no a priori reason however to assume that masses of objects behave as real numbers. The proposed experiment can be seen as a check to see if masses actually can be modeled by (a subset of the) real numbers.
