Is it true that all salt is sea salt

In a very real sense, yes, that is a true statement. Virtually all sodium chloride on earth was formed in the ocean. It’s a bit of a misnomer to consider “ocean” to be synonymous with “sea”; technically, and according to the National Oceanic and Atmospheric Administration there is a difference:

Many people use the terms "ocean" and "sea" interchangeably when speaking about the ocean, but there is a difference between the two terms when speaking of geography (the study of the Earth's surface). Seas are smaller than oceans and are usually located where the land and ocean meet. Typically, seas are partially enclosed by land.

Since many manufacturers of “sea salt” evaporate ocean water independent of any actual “sea”, I’ll use that small misnomer too, and equate “sea” to “ocean”.

Salt (sodium chloride) deposits were born in ancient oceans in a process that started over four billion years ago. At one point in time or another, almost all of the planet (exactly how much and when is still a matter of debate) has been under an ocean. Rivers flow into the oceans, bringing with them minerals from erosion including positively charged sodium. Volcanic eruptions from the ocean floor spew gasses into the ocean including negatively charged chlorine.

Salt lakes share a quality with oceans in that they don’t have an outflow, the volume of water is balanced by evaporation and rainfall. Water evaporates, but salt does not. So the salt created (and is still being created, but at a much slower rate) by the oceans concentrates in an ongoing process in salt lakes.

Salt mines are just dried up oceans.

So, marketing sea salt as special because it’s from the sea is a bit disingenuous. There are many reasons to choose one type of salt over another, but that it comes from the sea isn’t one of them. Other questions here address trace minerals found in certain mined salt, the addition of iodine in some table salt, crystal size and refinement methods; but the original source of all salt is the same.

There is a kind of fun exception: Some salt on earth comes from asteroids and meteorites. Garnish with that, Thomas Keller.