We don't.

Brute memorisation works for some people - me, in some cases - but frankly, I don't give a single solitary damn about dates or wars or names, and each one I have in my head is relevant because it impinges on some way in my study. Put another way, it's relevant because it matters.

Take my specialisation. I have exactly one year firmly fixed in my head, and that's 1346. That's the year that the city of Exeter signed a contract with its cathedral and St Nicholas Priory, dividing the water piped in from a spring outside the city so that one-third part went to the city, one-third went to the cathedral, and one-third to the priory. That's it. That's the only firm year I have fixed in my head. I also (technically) have 1000, because aqueduct construction in the Middle Ages picks up after that year, but that's more like a range, and there's a caveat to that - that's aqueducts in monasteries and palaces, with city aqueducts picking up in the 1100s and 1200s.

Similarly, Exeter was caught up in the Prayer Book Rebellion. I don't know when the Prayer Book Rebellion was - it's either the 1500s or the 1600s, don't know, don't care. The reason I don't remember is because Exeter was besieged once in each century, so I can't remember which century the Prayer Book Rebellion fell into. But it doesn't matter for me. What matters is that, in both sieges, the besieging parties dug up the parts of the aqueducts and cisterns that lay outside the city. Since these were made of lead they were melted down for use as shot.

History education below college level (and even at the undergraduate level, I am given to understand) is, at best, a hot mess. What is taught at school is not How Historians Do Business. Yeah, sure, some historians may be able to pull out a bunch of dates and names off the top of their head, but none of us is really concerned about that - we look at motivations, social trends, how life changed as a result of Event Y or Thing A, and such things like that. Any dates, wars, or names memorised is instead a side-effect of our study, mainly because we keep running into them so often. But again, it's a matter of relevance; once you poke a historian for a name, they'll also be able to tell you why that person mattered and what they did.

But frankly, there's no need to memorise dates, wars, or names. If you really have to, you can grab a timeline or a basic list of events off Wikipedia. Yes, we do use Wikipedia. It's pretty good with the basic details and it's readily available. It's not so hot on the detail level or when you start getting into historiography, but then, that's what historians do anyway.

tl;dr: uh guys when was the prayer book rebellion again?

I doubt the vast majority of historians memorize things like dates or names though I'm sure some historian actually do prefer at least for some things a more mnemonic approach. I don't know if it's different in other countries (I don't think so) but in my countries historian usually specialize in a chosen period of time / a specific country. So a medievist specialize in the history of European Middle Ages. Yes you have to remember a lot of things but you mostly have to remember the general context of the period of time you're studying and the most important facts and cultural features. Obviously some dates and figures are so important that it's basically mandatory to know them. An european medievist who doesn't know who Saint Louis is would probably never get a degree. It's ok if you don't always remember who exactly was the duke of a specific city in a specific year, but it's not ok if you don't know something like how did an abbey work, how did it evolve in time, what kind of relationship it had with secular power. Of course when an historian is doing speeches, public lessons, interview and things like that they would usually re study the topic so they can be more precise with names, dates and things like that. It does give the appearance that they're basically walking history books but it's just because of the continuous studying and researching that's behind such a profession.