No, metaphysical necessity is not coherent, and basically cannot be captured.
Kripke's efforts to revive necessity as a valid concept in philosophy explicitly used the method of providing examples that intuitively felt to philosophers that they involved a "necessity" relationship. The necessity is not a "logical" one, so Kripke invented a weaker form of "necessity" to try to characterize what was meant.
Why do this?
Analytic philosophy was the dominant movement within philosophy for the 20th century, and the most clear articulation of it was from Logical Positivism. LP used absolutist definitions to justify its approach to philosophy, which was primarily logic based, and which treated science as a favored stepchild of logic. LP also used these absolutist definitions to try to banish most other approaches to philosophy.
However, non-analytic philosophers pushed aback against the LP attack, and the pragmatic empiricists in particular basically broke the LP movement. Popper showed that science uses refutations, not confirmations, and that all of science is radically contingent. This breaks the "necessity" needed to apply logic to this world. Hume was referenced in that objects in our world are "bundles", not reducible to the same entity over time, hence "we cannot step into the same river twice". So for our world, A=/=A, and logic cannot be applied to it. And Quine argued that as we each have to make guesstimated inferences as to what any language term refers to, all language is a personal inference, and A=/=A between any two language speakers so language also cannot support logic.
Quine argued that this broke the analytic/synthetic distinction, and that all knowledge was synthetic -- which most philosophers considered a step too far in his reasoning, but the A=/=A of language does not require accepting Quine's total breakdown of the distinction. And for any logic to apply to our world, there was increasing recognition that "identity" of an object requires it have an "essence", but essentialism is an Idealist concept, and the vast majority of philosophers have adopted physicalism as a worldview.
The consequence of this pragmatic empiricist pushback against LP, is that the applicability of analytic philosophy to our world was in question. Most analytic philosophers tried to adopt a "descriptive" approach to identity, where identity is established by some sufficient fraction of the descriptive elements applying. Kripke appropriately found descriptivism to be far too loose to support analytics -- as The Ship of Theseus and part substitution thought problems readily reveal (he spells out a set of critiques of descriptivist identity in N&N, which most philosophers consider decisive). So he set out to challenge the contingency of our world, in order to salvage analyticity.Kripke therefore set out to find a place for "essences" in contemporary philosophy.
Kripke's primary method to do this, was to offer thought problems that appeal to philosophical intuitions of their being necessarily true.
Empirical critique of Kripke
Kripke bases his argument on a variety of thought problems of identity, which he claims are necessary:
Water is H2O.
Heat is molecular motion.
Gold is the element with atomic number 79.
Hesperus is Phosphorus
cats are animals
Samuel Clemens is Mark Twain
The first four of these all illustrate aspects of the same problem in Kripke's thinking, and that of most analytic philosophers. Kripke and analytic philosophers focus their study on MATH and LOGIC, not on science. There is a common assumption among analytic philosophers that even if events in our world are contingent, that there are unalterable LAWS OF PHYSICS behind them, and that those laws will fix the relationship between Water and H2O, and Gold and an elemental number.
BUT -- "laws" in science are not unalterable. They are not necessary, but are just regularities. They are the result of fundamental symmetries which were discovered in Noether's Theorem https://hackaday.com/2016/06/14/symmetry-for-dummies-noethers-theorem/. So -- "laws" are derivative, not primary. AND -- the symmetry that creates laws, is spontaneously broken, in every case, so our laws all have exceptions https://www.jstor.org/stable/41065.
Note, if neither laws nor symmetries are necessary, then the physical relation between things that Kripke lists cannot be necessary either. The intuitions of analytic philosophers on this question, are wrong.
Addressing his specific examples -- if the values of the constants of the Stannard Model of Quantum Mechanics were VERY slightly different, and per physics thinking, they are contingent, and COULD have been different, then H2O would not create the properties of water. Similarly, element 79 need not have produced the properties of Gold.
It would require some more complex change in the nature of physics for heat to not be the result of molecular motion, and it is hard enough for physicists to come up with tweaks to the Standard Model that even allow for any matter to appear at all, so an alternate physics speculation that produces a "heat-like" phenomenon that isn't molecular motion has not to my knowledge been developed, but there is no theoretical obstacle to this.
The details of orbital mechanics in our world would prevent two stable planets around the sun in Venus's orbit, and would not couple them to the Earth's rotation, hence in our world, we have discovered that Hesperus must be Phosphorus. But it would not require absurd tweaks to gravitation and our solar system structure to allow for a two planets to orbit in Venus's orbit and be stable for millennia, and some rotation rates of the earth could then lead to their being different. This identity too, is contingent, not necessary.
That cats are animals -- is subject to the problems of indefinite definition in our world. The margin cases for cats and animals are basically impossible to nail down, hence they are best understood as APPROXIMATE category terms, rather than logic categories. And that cats happen to be animals, was discovered by investigation -- it is possible in principle that mobile entities could be from other kingdoms. True, few plants are mobile, but they could be in principle, and the same with fungi. And these three kingdoms of multi-celled life HAPPEN to encompass the macro-scale life we have encountered, but there could in principle be lots of living things in the universe that do not fit any of our taxonomical categories, and cats could have been the first such discovery. That cats happened to fit into one of these three categories very well, does not make their fitting NECESSARY!
Kripke uses "modal reasoning" that resorts to "alternate worlds" thinking, but his "alternate worlds" are, in basically every one of these cases, insufficiently imaginative to actually encompass the alternate potential worlds considered by contingent science.
Samuel Clemems necessarily being Mark Twain was argued by Kripke through different means. He admits that Samuel Clemens may have had a different name, and may have been a very different person in "alternate worlds", and that somebody else could have written the Mark Twain stories. What Kripke argues is that the experience of selfhood is different for different people, and in these alternate worlds, that the "self" of our world's Samuel Clemens should be identified genetically, which could well be with other individuals who might be very different. Here Kripke has inapproprately gone reductionist, by coupling these other worlds to our Samuel Clemens by limiting him to the same parents and genetics. This reductionism does not fully capture our human intuition, or biological knowledge, as a few small genetic changes in Samuel Clemens would leave him VERY similar although not identical, such that a non-genetically identical Clemens could very plausibly have written the Mark Twain stories. And actually genetically identical Samuel Clemens in other worlds could have very different personalities and skills and may have been incapable of being the Mark Twain author. Most of our intuitions at that point would tend to lump that anyone who could authored the stories as "Mark Twain", and the genetically identical by non-author Clemens as not "Mark Twain" in any way. In this argument at any rate, Kripke failed to adhere either to essences OR to philosophic intuition.
So -- none of these examples are actually necessary, per the current conventions of scientific empiricism.
In additional discussion, Kripke tries to defend against Quine's point that A=/=A in word meanings between people, by arguing that people can use a Proper Noun properly (I.E. logically validly) without knowing all the details of the meaning of that noun. This argument relies upon their being an essence to the universe that the noun can refer to, without being fully cognizant of all aspects of it. The principle is plausible, provided that there is such an essence. How one gets an essence of "The Mississippi River" out of the bundle and variable nature of our world -- is still a problem for Kripke. He tried to put the burden on language, by assertion a "baptism" for such a term, and then learned usage through the community of speakers. BUT -- this still only justifies an approximate and fuzzy usage, not his "rigid designator" claim, and does not actually support valid analytic usage of any such proper name.
The pragmatic alternative
These empirical critiques, if appropriate, rebut metphysical necessity, and with it Kripke's justifications of analyticity. They leave our world entirely contingent, and analyticity inapplicable by its own standards to anything in our world, and to communication (language) between us.
This does not mean analyticity is useless. The pragmatic alternative is to accept that analyticity is not valid PER ITS OWN STANDARDS, but still can be useful, PER THE STANDARDS OF PRAGMATISM. There is a great deal of insight we can gain from applying analytic methods to our world, even if we cannot always trust its conclusions. We can communicate usefully, even without logical certainty of our shared meaning. Pragmatism does not demand necessity, nor certainty, only utility.