How does one interpret the terms "Linear" and "Congruential" as in a "Linear congruential RNG"?
I am used to linearity by $f(ax)=af(x)$. This does not seem to me to hold true in this case ($\bmod$). I have no idea how to interpret the congruential part.
Exactly What It Says On The Tin.
Let's break it down by looking at the definition. An LCG is any PRNG that takes the form
$$x_{k+1}=(ax_k+b)\bmod M$$
where $x_0,a$ and $b$ are some integer parameters, and $M$ is a large integer only slightly below the largest representable integer on the machine.
We can see where the name comes from (which, BTW, is due to D.H. Lehmer): "linear" is due to the fact that the quantity whose remainder we are taking is the result of a linear function ($ax+b$), and "congruential", since we are performing a congruence operation (modulo).