When using kosher salt one has to use more kosher salt than table salt to achieve the same level of saltiness. Does it mean that using kosher salt causes one to consume more salt (or sodium)?
Kosher salt is the same thing (though I'm pretty sure it's not iodized), but the crystals are less dense and larger. There's more air in each piece, and they don't pack together as tightly. That means you have to put a larger volume of kosher to get the same amount of actual sodium chloride (salt).
Think of it as the difference between ice cubes (regular salt) and snow (kosher). Both are water, but ice cubes are way denser than a scoop of snow. When melting it, you'll need more volume of snow to come out to the same amount of water as you'd get from a few ice cubes.
So no, it's not worse for you.
Kosher salt and table salt are equally "salty", however, kosher salt is less compact, which means that the same volume of salt will not be equally salty.
PepsiCo (owner of Lays chips) developed a new custom salt for potato chips, which allows them to use far less salt to achieve the same level of saltiness. This will allow them to cut sodium levels by 25%. So it's certainly true that the form of the salt has an affect on the amount of sodium you are consuming compared to the saltiness of the food. Some of the comments posted on this answer call into question whether you should care about sodium content. Assuming you should care, then it's worth considering it.