If you are going to think of a composite as a metal, think of a steel spring, the material is going to bend and return to its un-deflected position, until you over stress it, it is going to yield very little, then it will snap like a carrot. Once the plies are failing, you are into the "Carrot" phase.
B.E.

You are judged not by what you know, but by what you can do.

First ply failure with PMC almost always means some matrix failure but no fiber failure. Unlike yield the composite may return to its original shape. Any very small permanent deformation is likely to be from some plastic deformation of the resin, which a) may be very small and b) may be indiscernible due to the fibres springing back and forcing the resin back to very nearly the undeflected shape. If metallic yield is 0.2% permanent set (the usual material definition) it is often quite apparent on unloading.

Repeated FPF load on the composite may result in the resin damage propagating and merging with other resin damage. What happens as metal is repeatedly deformed to its yield stress is that there is theoretically no more plastic deformation but fatigue is of course quite likely with repeated appllication of Fty. How similar these effects are I'm not sure. Even carbon PMCs fatigue, with resin cracks merging and eventually some fibers breaking.

While prediction of last ply failure (fiber failure) is usually taken to be ultimate I'm not quite so sure of the FPF being considered as 'yield'. Some of the signs of yield may not be present but signs of FPF may include perceptible acoustic emissions.

I think considering a yield criterion for metal as having an equivalent in composite to be FPF might be misleading. For most steels and aluminums there will be quite a bit of margin between yield and ultimate. The difference between FPF and LPF will almost certainly be quite different from yield vs. ult in a metal. LPF vs. FPF will depend on the loading and layup even for the same material. And of course the difference between onset of metallic yield and ultimate failure varies massively depending on the alloy. Is onset of yield in a toolsteel like onset of yield in mild steel?

Perhaps use of analogy in engineering is of limited use in general?