First I will say that the OP's question and all the answers here (including the latest one from a minute ago) are great and I rate them with +1:) I will only supplement them with a few more extravagant but "thought-provoking" considerations...
"The question is why does the voltage drop across resistors of the same resistance vary from the first circuit to the second circuit? Does it have anything to do with current? Why does it happen? I am trying to find an intuitive explanation as to why it happens."
"What I really want is an answer as to why the voltage drop in the second circuit across each resistor is half, even though they have the same resistance as the one in the first circuit."
If you really want the voltage drops across resistors with the same resistance to be the same, I can offer you a solution - just replace the voltage sources with current sources. This is not just a joke but a very real circuit configuration that we can observe in some well-known electronic circuits (e.g., in the so-called "common-emitter stage with emitter degeneration" or "phase splitter").
But let's go back to the OP 1- and 2-resistor circuits powered by voltage sources and draw some interesting conclusions.
The first is that we may not be interested in the current flowing through the resistors and their resistance. In both circuits the voltage does not depend on either the current or the resistance. In the second circuit, the voltage drop across a resistor depends only on the ratio of its resistance to the total resistance.
A second interesting conclusion we can draw with respect to the Transistor's potentiometer. Although this is a variable resistor, when we rotate its wiper, we do not actually change anything - neither the resistance ... nor the current ... nor the voltage. We simply measure (choose) the voltage at one point on its internal resistive layer... but all other points have linearly decreasing voltages.
Of course, we can imagine that when rotating the wiper, the one partial resistance increases when the other decreases so their sum stays constant... and, as a result, the current is constant as well. We can see such "electronic potentiometers" in CMOS stages, current-feedback amplifers (CFA), etc.