I think this can best be answered not by looking more closely at the American Civil War but by taking a step back and looking at larger patterns in naming conventions across many civil wars in different parts of history. While it may not be always the case, in general, it is very common for the different sides of a civil war to be referred to by different names than that used for their collective nation. There are exceptions to this (as well as some murky areas between what counts as a civil war and what is just a revolt), but the general pattern holds true. As some examples:

When discussing the Spanish Civil War, the two sides are generally referred to as the Nationalists and the Republicans with neither being called the Spanish Army.

In the English Civil War, the two sides are generally referred to as the Royalists and the Parliamentarians with neither being called the English Army.

In Ceasar's Civil War (one of the largest and most famous of the very many Roman Civil Wars), the two sides are generally referred to as the Caesarians and the Pompeians with neither being called the Roman Army.

When looked at in this light, having neither side referred to as the American Army in the American Civil War fits the same naming pattern. This practice is used because, in all of these examples, both sides labeled themselves as the demonym previously used by the united nation. Regardless of your feelings towards ongoing issues stemming from the conflicts, it is a potential source of confusion when discussing the conflicts to use a name that could apply to either side to refer to just one side. Clarity of communication is important in this case, so most people default to using terms that are unambiguous.

As /u/Georgy_K_Zhukov points out, there are other terms that avoid the ambiguity while still clarifying some of the nuance that simply using "Union" and "Confederate" might muddle. However, note that with these suggested terms, the term "American Army" is still avoided due to the ambiguity of the term. However, US Army is encouraged. This is because while the Confederacy claimed to be American (their full name was the Confederate States of America) they never made a claim to be affiliated with the United States. This combined with the fact that many period documents refer to the Federal forces as "US" or "United States" makes it a term that is both unambiguous and period accurate.

Your suspicions are on the right track, but of course things are a little more complex than simply saying "It's the Lost Cause" and leaving at that, and while obviously we can't call them the same thing and "make it easiser to differentiate" is an obvious necessity, there are multiple options out there, so the specific one chosen inherantly says something.

At its most basic, we ought to start by simply noting that there is NO SUCH THING as truly neutral language. That isn't to say some choices aren't inherantly more controversial than others, and and many linguistic choices are so absolutely unremarkable in how broad and concrete a concensus we have on their application that diseciting them might seem an excersise in futility, but all the same, there is no such thing as truly neutral language.

This is particularly important when it comes to history, and the words that historians choose to describe what they are studying. We can see this play out all the time in public debates about what history means, and the Civil War is very much a long running, and still very current battleground where that happens, with debates about commemoration of the insurrectionists - or rebels, or Confederates, or Confederacy, or Boys in Butternut, or traitors, etc. - through statues, memorials, place names, public holidays, and countless other ways raging on.

Now, I am going to work somewhat backwards here as one thing I want to make clear is that so much of the terminology we use is unthinking! There is nothing inherantly wrong, for instance, with casually talking about "Union Soldiers". It is so frequent and so common that we just do it unthinkingly, as do many authors even. But I've literally had people respond with 'mind blown' responses simply by pointing out that the "Union Army" is the "American Army". It is one of my favorite, go to examples for what I mean in the above about language impacting perception. The simple act of calling it the American Army can signifcantly reframe perspective on the Civil War.

So now we get to the why. Well, if we go to the war years themselves, while "Union Forces" was not entirely unheard of, in reporting of the time, 'Union' far more frequently would be used to refer to the cause, and the most common term you would find for the soldiers fighting for Union would be Federals. Federal soldiers, the Federal Army, and so one being a common and frequent way to describe the boys in blue. As for their opponents, rebels would be quite frequent, and a variety of other terms too, but you would rarely find reporting in the areas not in rebellion that talked about the insurrectionist forces in a way that legitimized them, but rather contextualized them as forces in rebellion.

Fast forward to the immediate aftermath of the war, and this generally continued to hold true, but in the decades following, reconcilliation became an important goal. This played out in many ways, and much of it ended up giving certain levels of creedence to the rebel cause, and embracing the narrative of the Civil War as one which was a war of brother against brother, and while not quite legitimizing the Confederate cause, certainly valourizing it and allowing a certain level of infusion by the Lost Cause narrative into the national rememberance. As such, it simply couldn't be a war of insurrecitonists fighting the US Army. Again, while we can find 'Union' used during the war, this is a crital reason for why it would become the primary descriptor to ensure afterwards. It elided over just who were the legitimate forces of the legitimate, elected government of the United States. It also leaned into another critical factor in the valourizing which was to largely erase slavery from the Confederate cause, and African-Americans - who made up roughly 10 percent of American forces in the war! - from its memory.

Conventional wisdom about the war which was shaped in the last decades of the 19th century held powerful sway past the mid-century, and although there were always some scholars trying to push back against it, it was only in the 1960s and 1970s that we really would begin to see significant attempts to pushback against those narratives infused with Confederate apologia, but even then, some of the basic terminology - such as 'Union soldier' - had become so ingrained as to seem there to stay.

Its only really been in the last decade of so that we've begin to see the upcoming generation of Civil War scholars really begin to reevaluate just why we're using that term, and just what it conveys about the conflict - whether consciously or unconsciously. As often is the case these days, such discourse started on social media, but has by now become quite powerful, with many scholars starting to reconsider their terms, and even some academic presses debating how to handle it. This movement finally really broke through last year when the Army University Press announced its new guidelines on terminology to use in discussing the Civil War which can be found here, The guideliens focus on both sides of the conflict, in the first discussing briefly why their preference is "U.S. Army", "Federals", or "Federal Army" instead of "Union forces", and also discussing why they will continue to allow the use of "Confederacy" or "Confederate Army", as they don't believe that recognizing they had formed a 'political and social alliance' necessarily means recognizing its validity, since such an alliance can exist in rebellion.

They also, I would add, discourage the use of "North" and "South" - although recognize the occasional need for 'grammatical variety' - as allegiences and beliefs were hardly so neatly aligned as to be perfectly in lockstep with the nominal borders of those divisions.

As an historian who does find himself writing on the Civil War with some frequency, it is, on the whole, a very good policy in my mind, and one which I hope more Presses follow suit with in the future.

So hopefully this all has been at least a little enlightening for you, both in the why things were, and also as regards their potential future. The sum of it is that words mean something, and we often are fairly uncritical about the underlying specifics of what that, er, means. We aren't using "Union" to subtly equalize the two forces in the Civil War and portray it as a fight between two sides, both of which had reasonable causes worthy of historical commemoration by their respective decendents, but nevertheless the ways that we have historically talked about the Civil War - or the War of the Rebellion as it once was officially called - certainly has at times leaned heavily into that, conceeding much in the name of reconciliation. I also won't even say it is wrong to keep using "Union" now and then. I find myself slipping into it from time to time, and as the AUP themselves implied, it can get really damn tedious using the one, single term over and over so sometimes you just need to have some variety in your writing, but I do think it is important that we consider critically when, how, and why we're using it, and how it is more broadly contextualized.

(p.s. I do apologize for the likely excessive amount of typos. I'm in a hotel room typing this on my phone. I've been wanting someone to ask this question for some time as I have THOUGHTS on the topic, though, so damn if that impediment was going to stop me from jumping onto it)

While more can always be said, until then, you may be interested in this previous answer of mine to a nearly identical question.