Do kilns get hot enough to fire their own bricks? Kilns for firing ceramics are lined with firebrick, which are also ceramic. Note that heating elements are often placed in grooves in the walls. However, different ceramics have different firing temperatures, so it may be the case that the walls can only be fired at even higher temperatures. I would think that this is the case, as ceramics at firing temperature are often weak and not far from the point where they start to soften and deform under gravity.
Is the maximum safe operating temperature of most kilns below the temperature that could fire the kind of bricks in their walls? If so, are these bricks fired in a graphite furnace or other special ultra-high temperature process?
Firebrick are thermally stable for the temperature range they are specified. However , some materials may cause chemical changes in certain types of fire brick and develop lower melting phases. I am thinking of glazes dripping onto brick.  The normal Nichrome elements will not do this. Also silicon carbide elements will not deteriorate the brick.