In recipes, "table salt", "sea salt", and "koshering salt" often refer to different sized and shaped crystals. There is no question that when sprinkled on food they have different tastes, but that is due to the size and texture. Once dissolved there is no detectable difference.
It is possible to buy iodized fine-ground sea salt that is certified kosher. These terms are used literally on the packaging, as opposed to the way the terms are used by chefs.
In Do I Need to Use Kosher Salt? | Ask The Food Lab, J. Kenji López-Alt, the Chief culinary consultant of Serious Eats says:
While the differences in color are largely cosmetic, shape can have an effect on a salt's eating qualities. Chefs like using sea salts because they provide crunchy texture and a burst of salinity that adds interest to plated foods. They should be used exclusively for finishing dishes. Scattering on the tops of glazed loaves of bread before baking. Sprinkling over sliced perfectly cooked steak just before serving. Adding a touch of crunch to slivers of raw scallops. You get the picture. Fancy-pants food.
If you're using your fancy sea salt to cook with, on the other hand, you may as well replace your toilet paper with dollar bills, because you are flushing all of its good features down the toilet.
But really, all salt is sea salt. The only difference is whether it is extracted directly from sea water or mined from an ancient dried-up sea.
There might be very minor difference in mineral content, but not enough that anyone can taste the difference.
Where people might be able to taste a difference is when other chemicals are deliberately added to the salt when it is processed. In particular, iodine and anti-clumping agents are added in far greater proportions than any naturally occurring mineral.
A 1995 publication, Effect of Iodized Salt on the Colour and Taste of Food,
reports the results of a study commissioned by Unicef. This is its conclusion:
6. General conclusion
Studies on the influence of iodized salt on food quality have examined a variety of foods such as meat and dairy products, cheese, canned vegetables (tomato juice, green beans, sauerkraut, whole kernel sweet maize), bulk sauerkraut, white bread, baby foods, soup, pickled olives and potato chips. In none of the studies were adverse effects of iodine reported. There is one study which indicated that off-flavours may occur when iodized salt is used in conjunction with lemon flavouring but this is a minor problem which could be overcome by the highly advanced food industry which would bring such products on to the market. Another study in which iodine/ iodide mixtures were added at more than 100 times that normally used also showed that flavour could be affected. However such high iodine concentrations are unlikely to be achieved in practice. More importantly, there are undocumented reports that iodized salt produces changes in the colour and possibly texture of pickled vegetables. Because of the absence of data, further studies in this area are warranted.
There was a very slight correlation between iodine and taste, but not until concentrations that were at least 100 times what is added to table salt.