Skin irritation from Victorinox Fibrox handle
Black Fibrox is a trademarked brand name for Thermoplastic Elastomers, or TPEs.  This is a common plastic substance used to produce "grippy" plastics that feel rubberized, for a variety of applications.

Thing is, TPEs were largely invented as an allergy-free alternative to latex.  Studies have, to date, not turned up any naturally occuring TPE allergies in test subjects.  So you may be the first patient with a TPE allergy!  Lucky you.

To verify this, do you have any wearable electronics?  Items like the Fitbit and various smartwatches frequently use TPEs as band/covering material.  If you have one of these, and you are allergic to TPEs, you should be reacting to it as well.  Or go to your dermatologist, now that you know the substance you think you are reacting to, he can do targeted tests.

I can think of two short-term solutions for your knives.  One is to sell them any buy the versions with "rosewood" handles, which I believe are non-TPE.  The second would be to coat the TPE handles in something durable, like silicone.