Personally, I define job identity moreso over the gameplay than the job's "style", since that one tends to get muddled with how often roles have to share gearsets with other roles - often leading to looks that I loathe. If I had a dime for every time maiming sets looked the same as fending sets, making Dragoon look like a plummeting tincan instead of a jumping threat, I'd be rich. And while it's easy to say "summons define the summoner", plenty of reworks of various classes, in particular AST and SMN, have shown just how exchangable a lot of these things are.
Now, while I have all jobs in the game (minue BLU) at 90, I only play a handful in endgame content, so for some of these my idea of what the job aught to be might be ill-informed, but here goes nothing:
Paladin: The "gimmick oGCD" tank. Lots of stuff to press in between GCDs, along with some unique skills like cover and the ranged part of their rotation that don't really exist in other tanks really define its identity for me.
Warrior: The "I can heal" tank. Alongside with holmgang's short cooldown, the ability to self-sustain without losing uptime is really what defines Warrior to me.
Dark Knight: The Damage tank. In fights with lots of re-openers, their numbers are just insane, to the point where I'd consider it their defining feature.
Gunbreaker: The APM tank. Do you like cramming a lot of button presses into a very short amount of time? I am awful at it, so this sort of subsumes every other point of their identity for me.
Monk: The "micromanage" DPS. Is my DoT falling off? Is my buff falling off? How many chakra am I sitting on? Which form am I in? Oh god, there is so much to keep track of, please make it stop. MNK's pure business defines it for me.
Dragoon: The oGCD DPS. Absolutely zero brainpower required to think about GCD rotation. DoT reapplies itself automatically. One can focus entirely on the plenty oGCDs and fight mechanics in general, which is its unique property.
Ninja: The minigame DPS. I can just never separate anything NIN does from the mudra minigame you gotta play to go through it, mostly due to bad memories still lingering when it was a lot more ping reliant and I saw a lot of bunnies over my head.
Samurai: The "slow and steady" DPS. It feels the most "chill" out of all the melee to me. Its pacing is roughly the same throughout all the fight, with burst windows not feeling much busier than the rest. I'd also define it via its very high skill floor: It's very easy to be decent at Samurai. But it's quite tricky to be great since there's a lot of micro-optimisation, without it feeling overbearing like Monk.
Reaper: The two-mode DPS. I mostly define it over how different its two phases feel, with the rapid increase in GCD speed when transforming. It's gonna be interesting to see how much Viper will encroach upon that identity and how they'll differentiate it from Reaper.
Bard: Oof. Has seen so many reworks over the years, even I struggle to define an identity for it anymore. It seems like it should be the predestined support DPS, but that spot has been so thorougly taken by...
Dancer: The support DPS. I doubt I need to elaborate on this one. It's easily its most defining feature.
Machinist: The APM DPS. Basically GNB but for DPS. I define them in a very similar way, only for different roles.
Blackmage: The pre-planning optimisation DPS. While a better BLM might be able to do it on the fly, to me BLM is forever defined by how much pre-planning there needs to be done in order to play it efficiently. Knowing where attacks are potentially hitting ahead of time so you don't end up placing your laylines in the orange that's about to pop up, saving your triplecasts for the right moment, making proper use of Aetherial Manipulation to be able to keep casting for longer... In a way, it is the "strategy optimisation" DPS rather than a "rotation optimisation" DPS.
Summoner: Originally I had it as "DoTs DPS" but obviously that went out of the window with the rework. Now, I'd say its identity is the "mobile Caster", but honestly, that just makes it more similar to the ranged physical DPS. As a result, I'd say SMN currently has one of the weakest identities in the game gameplay wise.
Red Mage: The RNG mitigation DPS. Admittedly, this identity has become weaker and weaker with each expansion, as more tools to do just that have been added to the toolkit. Originally it felt like playing around whether certain spells procced or not, but that is slowly being drained away (ironically by adding quality of life to it, similar to AST - see below). As such, I doesn't have a very strong identity either, but still more than than SMN.
White Mage: The Big Healer. It doesn't have a strong gameplay identity save for its raw healing power. Originally I defined it more as the "MP healer" who never runs out of resources save for death, but as the other healers, AST in particular, had their MP gain buffed, this has faded.
Astrologian: The gimmick/support healer. Plenty of buttons that do unusual stuff compared to other healers, even if all the more interesting cards were taken away two reworks ago. And then, of course, the DPS-increasing cards. I guess you could also add the RNG to what defines their identity, for better or worse (definitely worse).
Sage: The resource proc/passive healer. When I think of Sage I mostly think of how proccing their resources works, requiring some forethought on when to use what skill to do so. Sadly, this means anywhere below endgame, their identity becomes just "passive healer" as Kardia goes brrr.
Scholar: This one I just don't play enough to have any opinion on. I guess the "complicated healer" since it looks super intimidating in complexity compared to the other three healers.