"True" is a tricky standard to apply to broad scale interpretations like this. This is basically an environment-determinist approach to thinking about these different civilizations (I almost wrote "two societies" but we are really talking about a lot of different societies over a long period of time) and their religions (and governments, which are linked to their religions). The basic argument is that the material conditions necessary for controlling these rivers, and thus ensuring high agricultural productivity, required certain types of social structures — e.g., hierarchical governments that would be able to muster the large-scale social resources necessary to build the irritation works and apportion land and calculate the sizes of plots prior to flooding and protect stockpiles of food and so on. This is basically the Wittfogel thesis, known as the "hydraulic hypothesis," and it is an argument about the origins of "the state" in general (that these kinds of civilizations are tied to water management).

So if you go into studying these places with this in mind, you will see things that confirm this view, for sure. But how would you know if it was false? That is, it's entirely possible to interpret aspects of these different religions as having to do with these hydraulic conditions... but couldn't one interpret almost anything this way, if you chose to? Religion is famously easy to interpret in essentially any way one chooses to, especially if you are given the liberty to pick and choose elements of it across thousands of years or hundreds of individual locations.

The Wittfogel thesis has, itself, not enjoyed continuous praise by archaeologists. For one thing, there is a chicken-and-egg problem, in that there is some evidence that many or most of the elements of these societies that Wittfogel attributed to irrigation management predated their use of large-scale irrigation techniques. That doesn't necessarily mean that Wittfogel's approach is totally bunk, in that you could imagine modifying it a bit so that it is not as dependent on being so directly (or at least, primarily) causal, but it does hint that there must be something more complicated going on here, and even from the beginning (Wittfogel published his work in 1957; major critiques were coming in by 1960) there have been arguments that the reality must be more multicausal than this. It has also been noted that there are evidence of societies that used large-scale irrigation that did not develop the hallmarks of what we think of as "the state." As such, archaeologists have been debating various forms of this for some time; it does not help that the thing that Wittfogel was trying to explain, "the state," is itself a contentious and arguably not super well-delineated concept from an archaeological or even political theoretical perspective (in the sense that the difference between a "pre-state" and "state"-like society is fuzzy).

Anyway, that is the origin of this idea. It is not inherently silly as something to teach, as a materialist approach to thinking about how the conditions of a place are marked on its culture, philosophy, religion, etc., are pretty standard in the social sciences (a legacy of Marx, which makes it sound scary to a lot of people, but one of the aspects of his work that pretty clearly has a lot of value). However whether it is literally true in the case of Egypt and Mesopotamia, well, it's hard to say.

A more recent critique of this approach from the David Wengrow and (the late) David Graeber (The Dawn of Everything: A New History of Humanity, 2021) argues pretty strongly against this kind of crude environmental determinism, pointing out many of the above issues and some others as well, such as examples of essentially "stateless" people coming together to make large-scale constructions (thus trying to refute the idea that one needs a hierarchical government to do this kind of work), and arguing instead that the arrow points in the opposite direction from Wittfogel: that cultural variations, like religion, led to different kinds of states and uses of the land, not the other way around. For Graeber and Wengrow, the form of the Egyptian state emerged from the religious idea that the dead got hungry and wanted to eat bread and drink beer, and thus they needed to create an agricultural empire to keep the dead satisfied. Their basic argument throughout the book is that archaeologists tend to underestimate how powerful and malleable human culture is, and that instead of seeing the state (or religion, or whatever) as just "emerging" out of material or social conditions (e.g., increased complexity caused by increased population), we should see states as deliberate cultural choices that were made. I want to quote this bit for you because it might seem like I am exaggerating or distorting their argument, and I don't think I am:

What preceded the First Dynasty, then, was not so much a lack of sovereign power as a superfluity of it: a surfeit of tiny kingdoms and miniature courts, always with a core of blood relatives and a motley collection of henchmen, wives, servants and assorted hangers-on. ... These kings give every sign of making grandiose, absolute, cosmological claims; but little sign of maintaining administrative or military control over their respective territories.

How do we get from here to the massive agrarian bureaucracy of later, dynastic times in Egypt? Part of the answer lies in a parallel process of change that archaeology also allows us to untangle, around the middle of the fourth millennium bc – we might imagine it as a kind of extended argument or debate about the responsibilities of the living to the dead. Do dead kings, like live ones, still need us to take care of them? Is this care different from the care accorded ordinary ancestors? Do ancestors get hungry? And if so, what exactly do they eat? For whatever reasons, the answer that gained traction across the Nile valley around 3500 bc was that ancestors do indeed get hungry, and what they required was something which, at that time, can only have been considered a rather exotic and perhaps luxurious form of food: leavened bread and fermented wheat beer, the pot-containers for which now start to become standard fixtures of well-appointed grave assemblages. It is no coincidence that arable wheat-farming – though long familiar in the valley and delta of the Nile – was only refined and intensified around this time, at least partly in response to the new demands of the dead.

The two processes – agronomic and ceremonial – were mutually reinforcing, and the social effects epochal. In effect, they led to the creation of what might be considered the world’s first peasantry. As in so many parts of the world initially favoured by Neolithic populations, the periodic flooding of the Nile had at first made permanent division of lands difficult; quite likely, it was not ecological circumstances but the social requirement to provide bread and beer on ceremonial occasions that allowed such divisions to become entrenched. This was not just a matter of access to sufficient quantities of arable land, but also the means to maintain ploughs and oxen – another introduction of the late fourth millennium BC. Families who found themselves unable to command such resources had to obtain beer and loaves elsewhere, creating networks of obligation and debt. Hence important class distinctions and dependencies did, in fact, begin to emerge, as a sizeable sector of Egypt’s population found itself deprived of the means to care independently for ancestors.

I liked the Graeber and Wengrow book on the whole, because it does do a good job of pointing out many issues with the Enlightenment teleological models of civilization and state-formation (and indeed, the problems with the term "the state"), and has lots of examples of richly complicated "non-state" cultures... but their bit about Egypt is easily the weakest part of the book, and it feels, I have to admit, very silly on the face of it. In general, their arguments against environmental determinism are not always quite as bad (they argue it is often rendered far too "mechanical" and underestimates the power of "culture," and give many examples of societies that went against the predictions of such theories), although I do feel that when "culture" is the ultimate answer, it is something of a non-answer, because it often just means, "it could be anything and there is no reason for it." I am not sure I buy that.