What is the sheathing underneath my brick exterior walls?
A better question would be, why is my sheathing getting wet?  

The grain of the sheathing suggests to me it is  Homosote  Homasote.  It should not be squishy, rather firm and resilent.

Something is allowing water past your brick barrier, and then, past what was vapor barrier (for the time, 15-30# tar embedded felt paper.  I would look up, overflowing gutters can get behind fascia boards and run down the inside of the bricks, soaking the sheathing if the vapor barrier was broken during the bricking (as sometimes happened, also mortar bridging can cause this).  In no case is it normal for the sheathing to be wet. Poor tuck pointing (especially around windows) can also allow water intrusion.

Once the water intrusion is corrected, all stud bays in the affected areas should be opened and the sheathing evaluated for viability.  Also, opening the cavity will allow faster drying out and possibly see where the intrusion is starting.  



As an aside, my 1941 house used a gypsum lath board for plaster.