Prior to Hurrem, an arrangement known as the Sultanate of Women prevented the Sultan from marrying and designating a wife. Instead, beautiful women from the kingdom were placed in the Sultan's harem and designated as "concubines," which made them potential mothers for the Sultan's children. The official reason was that many women were unable to bear sons that lived, so this arrangement was designed to keep the sultan from "committing" to one woman. Instead, the idea was that each concubine would have at most one son, and then the mother-son teams would fight it out for the sultanate after the current sultan died, with the mother of the successful sultan being the second most powerful person in the kingdom after the sultan himself. For instance, Suleiman had only one surviving son, Mustafa, by an earlier consort, Mahidevran.
As noted in your first paragraph, the rule had already been "frayed" in favor of a woman named Ayesa, who was married "in steps" first to the Sultan's brother, and later the Sultan himself. But it was Suleiman who abolished the rule permanently.
Hurrem caused Suleiman to revise this rule. (Details in the next paragraph.) The reasons are 1) she was extremely beautiful 2) she bore more than one surviving son 3) She won extreme favor with the sultan, to the point where he was willing to defy his own people, and 4) She was extremely astute politically, and capable of acting as Suleiman's "co-ruler," a role usually played by the sultan's mother. Finally, the sultan in question was "Suleiman the Magnificent" whose successes abroad gave him more "leeway" at home.
Because of the above, Suleiman broke or changed the conditions of his marriage, and made Hurrem a full "wife" (in the western sense of the word), elevating her above all other concubines in the harem. Her new title was Haseki Sultan, which gave her a monopoly on bearing potential heirs to the throne (at least after the execution of Mustafa). This effectively made her "second in command" to the sultan, a status not generally enjoyed by queens consort in Europe and Asia.