Colorful English equivalent for the French expression "mine de capitaine"

Solution 1:

Being [or sitting] “on top of the world" is often used to describe being (or the feeling of being) in a good place/position (like being a captain sitting in the captain’s chair?), but if you used it with “look/looking” I think the notion of “appearing wonderful, glorious, ecstatic” would replace the notion of feeling like that, as in this example from a headline the ‘The Daily Mail’:

Sam Frost looks on top of the world …

on top of the world
Fig. feeling wonderful; glorious; ecstatic.
‘Wow, I feel on top of the world.’
‘Since he got a new job, he's on top of the world.’

(sitting) on top of the world
Fig. feeling wonderful; glorious; ecstatic. [same examples as given above]
(from ‘McGraw-Hill Dictionary of American Idioms and Phrasal Verbs’ via ‘The Free Dictionary by Farlex’)

(sitting) on top of the world
in a happy position of advantage
‘With these figures and their other recent successes, the company is sitting on top of the world.’
(from ‘Cambridge Dictionary of American Idioms’ also from ‘The Free Dictionary by Farlex’)

For an option that resembles the French expression (structurally, at least, with its use of “have”) you could consider using the fairly literal translation of the first part of the French (“[avoir] une mine de …” = “[to have] the look of a …” ), which can be used idiomatically with just about any noun to imply whatever appearance is usually associated with the chosen noun.

Although “[to have] the look of a captain” would certainly be too literal of an equivalent and more importantly, would probably not imply much more than how captains are normally dressed, perhaps the appearance of some of the positive qualities mentioned in your 'Edit' could be implied with:
“[to have] the look of a winner.”

They must look healthy and energetic, the look of a winner

(quoted example, with emphasis added, from ‘On the Origin of Spin: (Or how Hollywood, the Ad Men and the World Wide Web became the Fifth Estate and created our image of power’ by Brendan Bruce, via “Google Books’)

Although not very colorful “The Look Of A Winner” is an idiomatic expression used to imply (sometimes incorrectly) that someone has (or appears to have) positive traits normally associated with “winners,” such as someone who successfully captains his/her own vessel to exotic vacation spots.
(used as the title of an article in ‘Scientific American’)

Solution 2:

I'm trying to think of a female equivalent... Anyway, that tall, tanned, handsome man that turns everyone's head is known as "a Greek god."

"Have you seen Mabel's new boyfriend?"

"No, but I heard he's good-looking."

"Good-looking..? He's a Greek god !"