X means to object to
X means to have some interest in and feelings about
In the negative, then,
- I don't mind
X = 'I have no objections to
- I don't care about
X = 'I have no interest in or feelings about
Both can function as assent to a request for permission
- I'm considering turning on the air conditioning.
What do you think?
- I don't mind ~ I don't care = 'Sure, go ahead'
Though, of course, they display different emotional attitudes.
Such negative statements seem extremely common in languages of the world for reassurance.
This is similar to the common English expressions
No problem(s), No trouble(s), No worries, Don't (you) worry ((your pretty little head) about it),
and the common Spanish expressions
De nada, No hay problema, No te preocupes (sobre eso), etc.