This planet was named after Krios, who ruled the ancient Kriosian Empire with his brother Valt. The brothers became divided over the love for a woman named Garuth, so eventually their empire fell. Krios kidnapped Garuth from her homeworld, Valt Minor, and took her to Krios Prime, which started a centuries-long war between the two worlds. (TNG: "The Perfect Mate")
By the 22nd century, the Kriosians were under the rule of the Sovereign Dynasty of Krios Prime, a family known on hundreds of worlds. The royal guardsmen of the Dynasty were known as the Sovereign Guard. Their leader was known as the First Monarch, a post which Kaitaama ascended to in 2153. (ENT: "Precious Cargo")
By the mid-24th century, after conquering Krios and suspending Kriosian sovereignty, the Klingons established a colony on this planet. It was the only Klingon colony on the border of Federation space. In 2367, the Federation was accused by the planet's governor, Vagh, of aiding the planet's rebels, who hoped to free the planet from Klingon rule. However, it was revealed that Klingon Ambassador Kell, in an attempt to split the Klingon-Federation alliance, was in league with the Romulans, who were secretly aiding the rebels with forged Starfleet weapons. (TNG: "The Mind's Eye")
In 2368, Krios and Valt Minor took their first steps towards ending an ongoing war as Kamala, the first female metamorph born on Krios since the mid-23rd century, was delivered for marriage to Chancellor Alrik of Valt Minor. (TNG: "The Perfect Mate")
This planet's quadrant of origin is inferred based on the position of its star system as seen in the star charts appearing in the Star Trek: Discovery episode "Magic to Make the Sanest Man Go Mad" and in the Star Trek: Picard episode "Maps and Legends". In the case of the latter, the Krios system is no longer depicted as being in Klingon space, with it being now in neutral space, and far from the border of Federation space.
It was never implied or otherwise indicated that the planet depicted in "The Perfect Mate" (and referenced in "Precious Cargo") was supposed to be the same as (or different from) the planet that was portrayed a year prior in "The Mind's Eye".
Though unseen in "Precious Cargo", the final reference to Krios was further complicated in a conversation between Trip Tucker and Kaitaama, when Tucker asks to visit (simply) Krios, causing Kaitaama to correct him by saying "Krios Prime". The inclusion of this trivia by the writers was unnecessary for advancing the plot, but suggested that there was more than one planet with this name, or, among other reasons, she was extremely proper about her planet's name, which easily follows the air of superiority she carried throughout the episode.
Furthermore, while both planets were called "Krios", in both dialogue and in script notes, visually, the planet's color changed, quite noticeably, between the two episodes in which the planet was shown. For that matter, "The Perfect Mate" also made no effort to reconcile, or altogether mention the prominent Klingon connection made in the planet's first appearance. An additional, and unrelated, script note from "The Mind's Eye" described the Kriosian Capital City as thus; "The architecture is simple, Spartan, very Klingon." The visual discrepancy between both episodes was reproduced for the remastered version of TNG.
On StarTrek.com, it was stated explicitly that there were two different planets and/or systems that shared the same name: "Krios".  In contrast, the Star Trek Encyclopedia (4th ed., vol. 1, p. 445) only listed a single entry for Krios. The planet was classified as a class M planet, which – by the 24th century – was located in Klingon space. The Kriosian system was named as a protectorate of the Klingon Empire. Furthermore, the Encyclopedia's entry for "Krios Prime" directs the reader to "Krios", indicating the authors treat "Krios Prime" and "Krios" as interchangeable.