This is a complex topic with a lot of room for miscommunication and misinterpretation. It is correct that slavery was not as pervasive in Persia Proper as in Greece, and wage labour is well-attested from the Persepolis tablets. However, in no way did the Great Kings attempt to abolish slavery in e.g. Babylon, Egypt or Ionia, and the use of Babylonian labourers in the construction of Dareios' estate at Susa suggests that they had no problems with making use of them. It is also likely that they made use of various forms of forced labour (e.g. corvee or "conscripted" labour or estate service in exchange for land) where the line between "slavery" and "labour" can get blurred much like the line between "tax" and "extortion" in other ancient contexts. An instructive discussion may be found in this topic, see especially the posts by /u/Aithiopika .

Was it in context of everyone being "slave", meaning just subordinate to the King?

This deserves to be addressed separately, and yes, the idea of everyone being the king's personal slave is present in Greek literature. The Greek word used is, I think, "doulos" in most cases, and this was used for slaves, bond-servants, household servants and so on. In Near Eastern ideology, it is the case that, as you correctly note, subjects were often referred to as "slaves" of the Great King, though in Achaemenid ideology this is most commonly seen as a mark of honor - that is, it is used to refer to his most loyal ministers (e.g., Dareios refers to his right hand, "Spear-Bearer" Gobryas, in this way). Assyrian ideology, which Achaemenid ideology drew heavily upon, focuses a lot on the physical subjugation of conquered peoples (bound in fetters, etc), and while this isn't exactly pervasive in what survives from Achaemenid art, it is certainly present particularly in the Behistun Inscription.

so it would make sense that somewhere there might be real slaves in the claimed but independent territory. I want to know if the mainstream empire practiced it.

It's not really possible to speak of a "mainstream empire" in the case of the Achaemenids, and their ideology makes a point of referring to it as 20-odd "lands" (dahya) over which the Great King's dominion (xshaca) extends. Rather, you have to look at regions separately. But there are attestations of the sale of slaves being tolerated around Persepolis, so it's not like there was a strong taboo against it.