This isn’t really a history question, and I think your first instinct about going to askphilosophy was correct. I’m not 100% sure if it’s breaking askhistory rules, so it’s possible it’ll be removed.

But the crux of what you’re getting at can in many ways be looked at through the lens of what Ludwig Wittgenstein wrote about in the early / mid 20th century. Groupings are inherently artificial, as they’re ultimately expressions of language rather than fact. This applies to history as much as anything else.

Take categorising animals as an example. Scientists have different classes of categorisation, but ultimately it’s an arbitrary line. At what point did Homo sapiens become Homo sapiens sapiens?

A history example might be defining a world war. We all know there were 2 world wars. But why is the Seven Years war not counted? It spanned the globe. Or why are the napoleonic conflicts not counted? They featured something very close to total war / total mobilisation. And why do we count the really quite separate wars of Japan / China as part of the same war between Germany and the European powers?

The reason is because it’s necessary to create these artificial categories to help begin to understand them. You are completely correct in identifying that the way it’s often taught broadly doesn’t hold up to proper scrutiny. The example you give of the fall of Rome is a great one. Western Rome’s fall was gradual. For those living through the period they would not have woken up one day in 476 and said “oh look Rome’s fallen”.

But just because the groupings are artificial doesn’t mean they’re exactly arbitrary. WW2 is said to have started in September 1939, with the German invasion of Poland. In Russia the “great patriotic war” as they called it began June 1941, when Germany attacked the USSR. But should we include the Spanish Civil War (1936) as the true beginning? Or Italy attacking Abyssinia? Or Japan invading China?

These events are absolutely important in contextualising WW2. But the usual interpretation of the beginning of WW2 being September 1939 is because that is the point you can see the most notable change in situation. There were conflicts before this date, eg Abyssinia or in China. And further conflict broke out after, eg with the invasion of Russia by the Nazis and the attack on Pearl Harbour. But it’s quite clear that the step taken on September 1939 was the most decisive towards an idea of a world war.

WW2 is an easy example, a more nuanced might be the one you give, about the Cold War. When did that start? Immediately after WW2? The closing of Berlin’s borders? The USSR developing nuclear weapons? It’s hard to point to a single moment. Not to mention the various ebbs and flows of it, with periods of heightened tension and periods of detente.

But we have the broad term “Cold War” basically because it’s helpful linguistic shorthand. Without the concept it would be difficult to give a broad explanation of events, and it would be difficult to build more nuanced understanding of the political / historical situation.

So to answer your question, what allow us to group things into historical periods or events is the fact that they are ultimately artificial, and we can do so if we choose to. The reason we choose to is to give us useful shorthand that allows much easier understanding of broad ideas, as well as to build a foundation that allows us to continue towards a more nuanced understanding if we desire.

This is what we call "periodization." The answer is that for pretty much any "event" you can and could have multiple start and end times, and one's choice of them is (at best) a deliberate argument about the nature of the event. So if you start World War II with Hitler invading Poland, you are saying something distinctly different happened at that moment than, say, Hitler annexing Austria or Mussolini attacking Ethiopia or Japan invading China or the Nazis aiding Franco in the Spanish Civil War. If you say that there was a World War I followed by an Interwar and then a World War II, you are saying something different than saying that the period of 1914-1945 was essentially one continuous conflict. And so on and so on. There aren't right or wrong answers here, necessarily, but every periodization draws some kinds of interpretive comparisons that emphasize some things and overlook other things; they are a way to say, "these times are similar" and "these times are different" and of course with pretty much all historical events there is continuity and there is discontinuity existing at the same time. What we do, when we do these things thoughtfully (and don't just adopt other peoples' periodization as gospel) is we try to be aware of what our periodizations do well and what they do poorly, and make it an explicit part of the argument. Even for "natural" periods (the span of a human life, the date some astronomical event happened, what have you), drawing attention to one period as important, over others, is its own form of argument.

I would not say that any of this is necessarily anachronistic; it is, however, interpretive, which is the essence of historical work. The place where these things get silly is when people believe the periodizations are all "natural" and then get upset or confused if people offer up alternatives, believing those people to be somehow rejecting a natural ordering of things, when in reality the original periodization was no more natural than the new one.