To give a theoretical perspective on this:
Going by your question, I assume the following interpretation of terms matches your intentions: A Democrat is somebody who is more likely to vote Democrat than Republican (and vice versa). I here introduce the notion of probability to account for effects of individual elections such as preferences for particular candidates, incumbency bonus, scandals, etc. Instead this is about general party preferences.
The US is a textbook example of Duverger’s law having a two-party system as the equilibrium state of an plurality voting system. In such a two-party system, there is a trend of parties (or candidates) to adjust their positions such that they have roughly equal chances of winning: If a party (or candidate) is currently on the losing side of things, it has to move towards their opponent to gain more voters; if a party (or candidate) is winning, it has no big incentive to move towards the opponent. By the way, something similar happens in multi-party systems, where there is a trend towards an equilibrium between the left- and right-leaning coalitions.
Now, there are other effects, such as regional variations, politicians following motivations other than winning (such as ideals), candidates and parties misjudging their current chances, etc.
However, most of these effects act on short time scales, while being a Republican or Democrat¹ changes more slowly and thus many of these effects are averaged out.
The thoughts going into this are very similar to the underlying assumptions of the median-voter theorem.
Essentially, my claim is an inversion concluding that the divide between two parties represents the median of the electorate (or more precisely, the point of equal chances to win an election).
Kramer’s A dynamical model of political equilibrium also seems to tie into this on a higher-dimensional political landscape.
There are studies showing that there is always a fixed distance between quantified political positions of the two main presidential candidates in the US (except for 2016), but I fail to find it right now.
So, in an plurality voting system, there is an equilibrium where both parties have equal chances of winning.
Now, let’s turn to the specifics of the US.
It is clearly close to that equilibrium, so if its voting system and regional structure were free from any biases (within the confines of plurality voting), we would expect that the electorate would be split¹ in half between the big parties.
However, US voting systems feature considerable biases towards Republicans, such as general effects of geographic distribution, rural low-population states having more political weight, gerrymandering, various forms of disenfranchisement (also see this question).
As a consequence, there must be more Democrats than Republicans¹ to achieve the aforementioned equilibrium.
¹ as defined in the first paragraph