In one sense, "Yes". A water softener using this principle just needs a source of Sodium ions, and Sodium Chloride aka salt is exactly what's called for in this application. They do literally just need salt.

But in another sense, "No". The people selling you table salt know it's for cooking and eating not to run a water softener, so they may tweak the product for these purposes. The product should be labelled, but we saw earlier that people don't read the fine print.

For example isn't it annoying when the little grains of salt stick together in a salt shaker? In some environments it's difficult to prevent plain salt from doing that, especially if it sits in the shaker for a few weeks because you (sensibly) don't use extra salt in most food. But an anti-caking agent reduces this problem while not making any difference (in a tiny quantity) to the taste or dietary effects of the salt. A water softener does not need anti-caking agents, so the manufacturer recommends using dishwasher salt.

Another example: Humans are supposed to have the element iodine in their diet. Some countries historically had a big problem with insufficient iodine, and so the "ordinary" table salt in those countries is laced with (a tiny amount of) iodine. Result: Fewer children with mental development problems. A water softener has no need for iodine, maybe it makes no difference, it clearly can't help, hence, the recommendation specifically for dishwasher salt.

Pretty much, the only difference is that dishwasher salt might dissolve easier in water, but it doesn't matter much, it's still just table salt.