I'd say it does say more about the person doing the treating. Even the lowliest, undignified person can be treated with dignity. There is no requirement for someone to actually have any dignity in order to be treated as such.

"treat someone with dignity" means "treat someone as if they have dignity". This describeds the manner of treating indirectly.

It's not the same sense of "with" as in "treat someone with medicine" or "treat someone with kid gloves".

It's the same structure as "to treat someone with respect" or "to treat someone with contempt". It expresses a relationship between the two people.

This Google NGram usage chart has plenty of written instances of must treat them with dignity, but it doesn't find must accord them dignity or must accord dignity to them. I think the question is just a peeve.

I’m voting to close this question because it looks like a peeve

@FumbleFingers; I assure you, it's not a "peeve". I really would like to know why people say it the way they do. I already know that the people who are saying it think they know what they mean, but I'm not sure it means that. I'm not offended by your downvote--or anyone's downvote, for that matter, so I'll wait patiently for someone to provide an answer that makes sense to me. Don

@NuclearHoagie: We'll need to agree to disagree, agreeably I hope. My point of view makes sense in an evanescent fashion. The concept is the kind that is hard to explain, but it makes sense--again, in an evanescent moment. At least that has been my experience. Don

@Barmar: Good point. Thanks for the comment. However, you'have changed the wording significantly. Don

"why?" questions like this are generally unanswerable. Language evolves organically. The best we can do is trace the history of usage, we can't explain why people started using words in a particular way.

If I treat you with respect, I am the one with respect for you. However, I cannot accord myself dignity in order to treat you with it...How would that even be possible? And also, I doubt dignified and dignity are linked the way you reckon they are.

@Barmar: I agree with you, at least partly. Saying that what people mean when they say things the way they do, seems to relegate meaning to non-meaning. Frankly, I understand perfectly what a person means when they say "I treat people with dignity." I would not correct or person or think less of them for saying it. Maybe my question is more appropriate for a philosophy website on Stack Exchange. Don

@Lambie: I'm not saying that I can accord myself dignity and then in turn treat you with dignity. Dignity is something a person has or possesses, and when I recognize it, I let the person know that I recognize it, particularly by the way I treat them. Again, as I've said to other commenters, the meaning of the phrase is kind of evanescent. Once you--really, I--have it, it evaporates. Don

Unfortunately, that's how language works. Nothing has any inherent meaning, it's all defined by how people use it. People say "literally" when they mean "not literally". Yet we understand it.

@StuartF, 'A treats B with respect' has the same grammatical structure as 'A treats B with dignity', but different logical structure: the respect in the former case is A's respect, but the dignity in the latter one is B's dignity. The question, in effect, seeks an explanation of the difference between the ways in which the language handles these cases.

You are not understanding me. If I say: I treat you with respect, I give it to you. If I treat you with dignity, dignity is not something I can give you. jsw29 says it better than I do.

@jsw29: I understand what you are saying--I think! If you turn your comment into an answer, I'll likely upvote your answer. Don

@Lambie: OK, I think I understand the way you re-phrased your last comment. Yeah, I think I agree with jsw29's comment. Don

For me, the difference is in the semantics. The terms respect and dignity don't work the same way.

@rhetorician, I thought I was only reformulating your question, rather than answering it. Going by syntax alone, one would reasonably expect 'A treats B with dignity' to work analogously to 'A treats B with respect' or for that matter 'A treats B with medicine'. But that's not how the phrase is actually used, and I think you are quite justified in being puzzled by that, and seeking an explanation of it. I don't have an explanation, though.