I would put it this way - in my way of thinking and cultural assumptions, any given relationship, close or not, one has a sense of whether asking favors is possible, and what sort are reasonable. Or else a sense of uncertainty about it.

The statement "let me know if you need anything" has a different meaning depending on what it was reasonable to ask already.

It adds no information, due to the meaning of "anything" being entirely contextual.

If I know that they know that the relationship is fairly close, affirming it is a nice thing to do, especially if I'm feeling badly.

But if the relationship is not so close and/or I'm uncertain whether it's appropriate to ask a favor, I can imagine feeling antagonized by the form of an offer which is not really an offer or clarification of where we stand.

If there is ambiguity, there remains ambiguity, is how I see it.

> The statement "let me know if you need anything" has a different meaning depending on what it was reasonable to ask already.

That’s the purpose of the statement. The person you’re saying it to should have a vague idea of what is a reasonable ask depending on relationship, and if not, they will just get denied. But it is genuine.

If I’m at a store, and my coworker asks me to grab a pack of gum, I’ll do it. If they ask me to go to a store 15 min away for something, I’m going to say no, but they probably would not ask me that anyway. But if a close family member asks, I would go to the different store.