What? The sleeping Zelda in Zelda II is a different Zelda from the one in the first game? This is news to me...
Yeah in game it's not specifically stated that it's a different Zelda but it is in the manual. Originally Sleeping Zelda was supposed to be the first ever Zelda in the series but that was retconned pretty easily once ALttP was made and as we know now the very original was Skyward Sword Zelda. Here's a brief little summary of The Tragedy of Princess Zelda I based on what Hyrule Historia gave us.
According to a legend passed down within Impa's bloodline, well before the events of The Legend of Zelda, Hyrule was a prosperous land ruled by a king who had a daughter, Princess Zelda, and an elder son, the prince. After the king passed away, the prince sought the pieces of Triforce that he did not inherit, but was unable to find them. A magician close to the king informed the prince that the king had told his sister something about the Triforce. The prince questioned her, but she did not say anything, even after being threatened by an eternal sleep by the magician. In his anger, the magician used a spell to put the princess into an eternal sleep, and died when the spell was completed.
Overcome with grief, the prince sealed his sleeping sister in the North Castle and ordered that all female descendants of the Hylian monarchy are to be named Zelda. While the princesses before this incident were named Zelda in honor of the original Zelda from Skyward Sword, this marked the beginning of Zelda being named so by law, rather than tradition.
To break the curse a hero must possess the complete Triforce, but obtaining the Triforce of Courage would prove difficult as the king of ages ago had hidden it within the Great Palace. Only one who proved themselves worthy would obtain it. In the end, Link obtained the Triforce of Courage and awoke the sleeping princess from her slumber.
Keep in mind this is indicated to happen at least century or more before Zelda 1 takes place which shows how the slowly Hyrule truly declined up until Zelda 1 and Zelda 2 AoL.
According to the official artwork as well, the new king (Sleeping Zelda's elder brother) was even threatening villagers with a whip kids and women alike. One of the reasons Zelda 2 is considered dark is due to this twisted backstory that lead up to it.