I've been thinking about Train's wise argument, and it's hit me:
The format that Cube is closest to is EDH, since it's all about singleton decks that are more than the sum of their parts (with some bombs thrown in to the mix).
The reason why I don't think Cube feels like normal Constructed is because, to me at least, Constructed is all about playing best-of-three matches against a meta with fine-tuned engines and sideboards (BO1 Constructed is basically just kitchen table, except you purely play with assholes instead of your friends). Cube doesn't work like that, at least partially because a common design goal is to prevent a meta from arising (because we avoid stale gameplay).
In other words, the deck you get out of drafting a Cube might feel Constructed-y, but the experience of playing with those decks won't be.
I like what you're saying overall, but I think it could be expanded upon further.
I think building a Cube can feel like building an EDH deck: you're trying to shape whatever you want the deck//Cube to be into a specific outcome. If I want a spells matter archetype in my Cube, I'm trying to include a bunch of cards that can interact on that space and play to the overall strategy I am looking to apply to that deck. Likewise, if I'm building a
spells matter commander, I'm also trying to select a group of cards that play well with the theme I've chosen. The big difference is that Commander decks get the ability to go super deep on a single theme, including cards that are hyper-specific to one archetype or require the Commander to function, while Cube cards need to have a broader appeal than the specific deck they may be included to support. This is why
doomsday is a snap include in every Kess commander deck, but it goes in only a small minority of Cubes with storm or spells as themes: it's very good at what it does, but it's very narrow.
Gameplay-wise, though, I don't think Cube and EDH are particularly analogous. The differences in nature between a 1v1 20 life format and a 1v1v1v1 40 life format select for different cards in viability for inclusion. Cards like
Tarmogoyf and
Lightning Bolt end up being really powerful Cube inclusions in most contexts because their rate is so good in a game where you only need to hit the opponent for 20 damage. But they're not particularly good in a game where you have to do 6 times as much damage to multiple different players in order to win. Additionally, the higher life totals in Commander lead to games being a lot longer (except for at tables where everyone is trying to win with an early-game combo), making an entire suite of cards that are usually awful or incredibly niche in Cube into format-defining staples.
But that still doesn't give us an answer for what Cube is actually like. While I agree that Cube doesn't always feel like normal constructed, I don't think this is because of the metagame. Every format is going to have a metagame revolving around the best deck or a small set of best decks, and trying to stop this from happening is basically impossible. I think most Cube designers try to stop any one deck from pulling too far ahead of the others, but that doesn't change the fact that something is always going to rise to the top, much like in actual constructed. However, metagaming as a player in a Cube format looks very different than in a constructed tournament. In tournament constructed, you start your day with a refined deck and a sideboard full of tools to help you deal with bad matches. You can bring your
Leylines of the Void for Dredge,
Shatterstorms for affinity, and
Pithing Needles for those pesky
Borborygmos (just make sure you name
Borborygmos Enraged and not
Borborygmos, lest you incur the wrath of the
rules lawyers). If you guess wrong about what you'll face, you get hurt, but you can adjust to that before the tournament. In Cube, you have no idea what your deck is going to look like or what you're going to face until you sit down and draft your deck. Therefore, your metagaming decisions have to come during the draft while you're still actively shaping your deck. For example, if you think the format is going to be full of aggro decks, you're going to want to try and pick up that
Nyxfleece Ram in your slow deck. However, you also have to worry about building the best deck possible with the cards you can draft.
Ultimately, I think this means that Cube ends up being the closest to the classic "60 cards I own: the format" experience as opposed to anything else. While there is still a lot of strategies involved in deck construction and picking sideboard pieces, you're less likely to face a specific type of opponent than you would be in a tournament setting. The big difference is that instead of people bringing their own random piles of 60 cards to play, they're drafting from a single pool. This means that players will be building their decks from a common set of cards on a roughly even power level, so a player's skill will be a larger factor in who wins as opposed to deck power level or financial status.
From my experience, casual group power band and finances were driving factors behind my wanting to build a Cube in the first place: I wanted to play casual constructed with my friends, but there was not a consistent power band that everyone was building and playing in. There were kids with decks that basically looked like 60-card versions of recent draft archetypes, and there were kids who had 4 copies of
every mythic baneslayer that was unbeatable without interaction. Someone even had a
Blistercoil Weird storm deck that they insisted on playing against literally everyone, even though only a few people found it interesting to play against. For my part, I got really bored after a year at Scout Camp, where I went 36-0 undefeated with Slivers, and realized everyone would get more out of the game with a Cube. My tangent here may seem a little random, but the more that I think about it, I feel that it's an important part of understanding my perspective on Cube.