Why are meaning and caring synonymous in most peoples mind?
By 'meaning', do you refer to 'meaningful life' (pervasive feeling of satisfaction), or semiotic reference? And by 'caring', do you refer to 'compassion for others', 'involvement' or 'motivational relevance'?
Well, to be precise, they relate to negation of meaning with the negation of caring. I think caring for the purpose of this discussion any of the three will work for the moment at least. With respect to meaning, I'm definitely not referring to meaningful life, more the semiotic sense.
I also suspect that we need to define "most people".
I think that most people have no opinion on semiotics and have never reflected on semantics. In this, they have the same unadultered relationship to 'caring' as all the other mammals, they chew what the universe throws at them.
If by most people, you mean "most people with two to ten semesters of philosophy education", or a subset of the intersection of the sets of people we both know plus the editor of Scientific American, which happens to be a very similar thing, there are probably three lines of argument that amount to what you initially propose:
1. Dread of giving up realism. The main argument is that dropping the understanding of meaning as reference to external facts would kill realism (here I agree), which would open up a hell of unconstructive skepticism and solipsism and should therefore be avoided on common sense grounds, and nihilism (i.e. not caring) would happen to be the (undesirable) normative angle of that hell. Of course, this is a naturalistic fallacy, but since the dreaders are looking for arguments in support of realism, not against it, they won't check very hard.
This is quite similar to the famously dumb "atheism destroys morality" argument, and could probably be countered in the same way (i.e. by collecting hard data on the nihilistic tendencies of non-realists, while controlling for the annoyance caused to them by stupid realists).
2. The intersubjectivity argument: If meaning is the result of a process of social agreements, which bootstraps further social agreements, then dropping meaning would impair higher level sociality. If caring involves higher levels of sociality, it might go down the drain. The argument implies that caring is the result of high-level social cognition (which is at least a little debatable), and that high-level social cognition either depends on our choice of philosophical premises (which is absurd) or which enforces these premises retroactively (because we obviously perform high-level social cognition, and thus all its prerequisites must be given).
3. The pragmatist stance: Motivational relevance gives rise to our constructivist exploration of our environment. If I turn off the motivational system, I might stop looking for meaning. If I take an entirely pragmatist or performative stance towards meaning (i.e. meaning is what I make it to be), then meaning would entail motivation. The inverse of the argument would be quite different (but similar on the surface): If I stop attaching meanings, I lose the object of my motivation, and thus motivation itself becomes unsustainable.