An integral kernel is, of course, an integrable generalization $K(x,y)$ of a matrix $M_{j,k}$. You could very loosely call this a "kernel" in the sense of the "core" of the formula for a integral linear operator. For comparison, Wiktionary tells me that in German, a Kerngehäuse is an apple core, while Kernphysik is nuclear physics.

But mainly, I think that the two uses of kernel, one for the null space and one for an integration matrix, is just a terrible collision of terminology that became standard by accident. It's nice when a mathematical term is an inspired metaphor or neologism. For instance the word "spectrum" for the set of eigenvalues of an operator was not just inspired, but also prescient and profound. (As I understand it, the term was chosen by mathematicians, by analogy with spectral lines in chemistry, shortly *before* the development of quantum mechanics.) But sometimes we're just unlucky, or maybe collectively stupid.

As Jan Kolar points out, the kernel of an operating system is a third metaphorical use of the word that makes vastly more sense.