Ambiguity in the definition of entropy

Entropy is a property of a macrostate, not a system. So $\Omega$ is the number of microstates that correspond to the macrostate in question.

Putting aside quantization, it might appear that there are an infinite number of microstates, and thus the entropy is infinite, but for any level of resolution, the number is finite. And changing the level of resolution simply multiplies the number of microstates by a constant amount. Since it is almost always the change in entropy, not the absolute entropy, that is considered, and we're taking the log of $\Omega$, it actually doesn't matter if the definition of S is ambiguous up to a constant multiplicative factor, as that will cancel out when we take dS. So with a little hand waving (aka "normalization"), we can ignore the apparent infinity of entropy.

Entropy logarithmically measure of the number of microscopic states corresponding to some specific macroscopically-observable state, not the system as a whole. Put another way: systems that have not yet found their equilibrium state, when left alone, increase their entropy. This would not be possible if the system had the same entropy for all macrostates.

Indeed, the driving principle of entropy in modern stat-mech says that we have some uncertainty about the underlying microscopic state of the system and that from a certain perspective (basically, the one where every macroscopic quantity we can determine is conserved) we can treat nature as simply choosing a microstate uniformly at random. (We have to tread carefully about what exactly uniformly means here but an “obvious” choice seems to replicate certain nice features, like that metals will have specific heats that look like $3R$ where $R$ is the gas constant—a result that I want to say is due to Einstein but I am not 100% sure.)

As a result of this principle of nature picking microstates at random, our equilibrium state is the macrostate which contains the most microstates, and our regression to equilibrium is a process of macrostates getting larger and larger.