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Depends. Hidden Depths is a fairly general trope. The reasons matter. If the character is so private because of shyness, I'd say Socially Awkward Hero. If they think they're above such silly things as socialising, that's Aloof Ally. It's also quite possible the character simply Hates Small Talk and/or treats whatever they're doing in the group and thinks Being Personal Isn't Professional. If none of these works, try browsing Introversion Tropes
Yeah, that's the problem: There is no actual stated reason.
The character in question is Ishiro Serizawa from the MonsterVerse films. All his character profile
◊ from Godzilla: King of the Monsters (2019) says is that he's "a man of quiet reserve" and is "fiercely private". Nothing really on why he's like that, but he just is.
Well, the character page you linked mentions some useful tropes: The Stoic (that might be simply his basic personality or a result of Heroic BSoD, or both), Married to the Job (only cares for his work), Heroic BSoD (bad happened and now he's a Shell-Shocked Veteran with Thousand-Yard Stare).
Not really, this trope isn't about lack of relationships, but downright paranoia about them watching. Also, we have such a trope? Wow.
A main or supporting character is very private about their private life amongst their peers, to the point where those around them — including those close to them — know almost nothing about them.
Not sure if Hidden Depths could work.
Edited by Henshin4Life