It's originally jargon but became increasingly absorbed into general speech and, as often happens when jargon is absorbed into general language, increasingly generalised from its original specialised meaning.
In the 1930s there was a non-denominational Christian organisation called "The Oxford Group".
They had a belief that it was necessary for spiritual healing and growth, that one tell others both about one's sins, and also experiences where one felt God had helped or made His presence felt in one's life.
Because this process was reciprocal in the group, it was called sharing as it matched the sense of that word of people partaking in something together.
This group went on to become "Moral Re-Armament" and later "Initiatives of Change", while a break-away group "Alcoholics Anonymous" became much larger and led to the formation of several related groups. Both of those groups moved away from their originally Christian-only origins in different ways.
Alcoholics Anonymous in particular, increasingly understood sharing as more psychological and less spiritual than the Oxford Group had (though still partly spiritual).
This then influenced the language used in the context of group therapy.
From there, it moved into a more general pop-psychological meaning, within a much wider population; to share was to tell people about your life and experiences, particularly those matters that one would generally "keep to oneself". (Note how the metaphor of "keeping" something "to oneself" is pretty antonymous to that of "sharing", though keep of secrets or one's own counsel is much older).
And from there, it became more and more generalised, to the point where one can share just about any information or opinion, and it is indeed used as close to a synonym for said as you say.
Is this correct?
Ah, well, "correct" is tricky.
It's certainly used by modern speakers of the language in this sense, and understood by many. So it's correct as far as that goes.
There would still be some that would expect it to be used only for something revelatory in some way, though different people would have different opinions as to just how revelatory and what sort of revelations were covered. So it's incorrect in so far as it doesn't match that expectation.
I would not use it in either of the examples you give, and find them poor writing, but not bizarrely so; I'd understand what was meant.
More generally, it can be used as a said-bookism where writers scan a page, find that said is heavily repeated and then strive to replace some of those said s with anything else they can fit in. The term comes because one could at one time buy "Said Books" which contained synonyms of the word said to aid the would-be writer in this way.
But since readers actually read prose, rather than scan them to see if there's repetition, and since the tags said and ask have a low impact while other possibilities more often stand out, the effect is very often a distracting one.
If in doubt, use said.