I just read an interesting theory and re-watched the part in the finale where Din Jaren offers to yield the Darksaber to Bo Katan.
Gideon specifically says "In order to wield the Darksaber again..." Implying that Bo Katan disgraced herself by losing it to Gideon before.
So if this is true, a worthy warrior could accept a yielded Darksaber, but Bo Katan has to earn it back since it was taken from her.
So, the Darksaber shows up in Seasons 3 and 4 of Star Wars Rebels.
In S3E11, "Visions and Voices" Sabine Wren finds the Darksaber in Darth Maul's hideout on Dathomir. She doesn't fight Maul for it, it is being displayed as a trophy.
In S3E15, "Trials of the Darksaber" Sabine is trained to fight with the Darksaber by Kanan Jarrus and Ezra Bridger.
In S3E16, "Legacy of Mandalore" Sabine properly wins the Darksaber by defeating Gar Saxon in a duel. Gar Saxon is wielding the Darksaber, given to him by Sabine's mother Ursa Wren, and Sabine is wielding Ezra Bridger's Jedi lightsaber.
In S4E2, part two of "Heroes of Mandalore" Sabine Wren passes the Darksaber to Bo-Katan, no duel, in the presence of the clan leaders, anointing her as the proper ruler of Mandalore.
How does this impact The Mandalorian? My guess is that Mando can't give the Darksaber to Bo-Katan, because he's not a proper Mandalorian. If he was a full-fledged Mandalorian, having won it in combat from Moff Gideon, he would be able to pass it to Bo-Katan and make her the rightful ruler.
Note also that we do not know how Moff Gideon came into possession of the Darksaber, or how Bo-Katan lost it. It is possible that Bo-Katan lost the Darksaber on a technicality, because it was given to her and not won. That would explain why she insists that it must be won in combat now. (I find that unlikely: in the Star Wars Rebels episode, everyone present accepts Bo-Katan as the proper wielder immediately and without question.)
It is also possible that Bo-Katan feels she must once again prove herself worthy before she can accept the Darksaber.