Of course one could say it is just different people valuing different things and that is that, but I think the multiple-number hypothesis is better
I don't think you make it clear as to why it actually is better. Or why it can't be combined with the different-values interpretation.
Going by Yvain's metaphor of seeing status as a ladder with different rungs it's possible that there are multiple ladders and that different people care about different ladders.
Looked at generally, status is pretty clearly contextual (different ranking for different topics or evaluations). I don't think your idea of a scalar for each context/topic covers all the complexity, though. Even within a given context, it can be multidimensional in that people can use partial orderings of how they think of themselves and you distinct from how they think others think of you.
My expectation is that judgment of others and use of reputation are insanely complex in humans - it's arguably the primary driver for recent (tens of thousands of years) evolution, and I doubt it can be summarized using a small vector.
I suspect if anything it's somehow both, where rankings on each of the various ladders are correlated enough with each other that you can find a "general factor of status", like psychometric g. Which, much like psychometric g (and probably more so) would be debatably a real thing, although I confess I don't know enough statistics to understand those debates.
Like, confidence is high-status, but it's not just high-status on its own, it also lets you "sell" whatever other status you have, taking you up those ladders. And of course there's the halo effect - if you're high-status on one ladder, people will attribute you with higher status on the other ladders they care about than if you weren't.
I am reading The other kind of status and it seems to me that status is seen as a single number, either objective, or in the eyes of other people in the group, or your own, either ordinal or cardinal, but at the end of the day you can say your status is 67 points or 12th in rank. And I think it is not actually the case! A few examples why it is more complicated:
Intimidation, power, authority
People behave in a respectful, deferential, submissive way to people they are afraid of, be that their personal scariness or power and authority. However this is not genuine respect. (Well, it is hard to say exactly - I would say for most of you it is not so, but OTOH there are people out there who like strength or authority so much that they truly respect those who can intimidate them, because they too would like to be intimidating people. Let's say it is not genuine respect in all cases.) If your neighbor is a cop and people behave with him extra tactfully because if he gets pissed off he may find an excuse for an arrest, is that status? Better example: crimelords, The Godfather (by "normal" people, not their fellow criminals).
The opposite: the purely moral status
People who are very, very good, and their goodness also means they are very meek, they are very much the kind of people who would not hurt anyone not even in self defense, and it is obviously showing - they get a strange kind of respect. Many people genuinely treat them with respect, but somehow it lacks certain aspects of the respect a high-ranking businessman gets, somehow it seems if people are so obviously harmless, the respect has less depth.
Most common status
I think most common cases of status have elements of both. To be high status you need to have power - not necessarily in the social-political sense, but in the sense of "the ability to affect things". For example, a good example is being very intelligent and learned. It is a kind of power. And you need to use that power in ways we generally morally approve of, for we don't really respect a criminal mastermind. But you also need to have a bit of an intimidation potential too, you should not look too harmless, of course you don't need to behave in intimidating ways, but still if people think "wow, I would not want such a smart person as my enemy, I could get a check-mate", that gives more depth to the respect. Perhaps it is better - less disturbing - if you call it not intimidation potential, but _ally potential_: if someone else would want to hurt you, does this person have anything to assist you in the conflict? Anything could mean intelligence, knowledge, social influence, charisma, political position, physical strength...
I dislike made-up evo-psy as much as everybody else, but this sort of makes sense in an ancestral environment. We respect people who are useful allies, tribe members, who have power i.e. abilitites or resources usable in affecting the world, but what makes them useful allies also makes them dangerous as potential enemies so there is also a bit of an intimidation potential as well, and generally we want them to use these abilities or resources for the tribe, not against it, which is probably where morality comes from.
But that is only the beginning
In the example above, status is not one number but two: power status 43, morality status 51. This alone demonstrates the problem with the single-number approach. However there can be so many numbers... I have seen very, very confusing and ambiguous status-setups in my life that probably came from many numbers.
- For example, some people assign high status to people who wear business suits and their female equivalents because it suggests a powerful social position, but also some people (young-ish) were more like "Ah, so you work. How boring. Worky worky working bee tehehee. Why aren't you a rich playboy or gangster who does not need to work?" So I saw a kind of a wants-to-work vs. must-work split here or I am not even sure exactly what.
- I saw people who were generally materialistic and yet valued wearing designer clothes more than driving an expensive car in China-Mart clothes, so apparently they assigned a number to style and a number to wealth and it interacted in non-obvious ways.
- Or simply at school - it was not-obvious, whether the students with good grades had higher status, or those who considered it a romantic rebellion against authority to not study and not write tests and not answer teacher questions. Many kids envied the courage of the second group but were still afraid of punishment and studied conscientously anyway and the funniest part was that in trying to satisfy both goals, they studied conscientously, got good grades, then lied about it and boasted they did not study at all and got that good grade purely on luck or smarts! Because studying was seen like being a teacher boot licker, almost as bad as a snitch... but of course getting and admittance letter into a university of law (= "Wow Rob is gonna be a rich lawyer!!") made him a hero so both studying and not-studying conferred status!
- Still school, easier example: in the breaks, being funny and entertaining was valued. In the phys ed class, skill, as we played a lot of ball sports (and it was not considered being a teacher's pet to be good at it), high skilled players were respected. The hierarchy visibly changed in the break before and after phys ed class. This is a fairly clear example of status consisting of multiple numbers, like Humor 43, Skill 71.
Of course one could say it is just different people valuing different things and that is that, but I think the multiple-number hypothesis is better: the same people valuing other people in different ways, as in the very first example (intimidated respect to the crime boss or policeman, respect but without depth to the moral saint), or valuing other people differently in different circumstances...