Since he's no longer a Mandalorian, the armorer may be giving him something of a brief pass, since he's likely to head to Mandalore and try to atone. In The Book of Boba Fett episode 5, where the beskar spear is melted down, and in The Mandalorian season 2, where he takes the suit from, this is established.
Since he's learned that the Way of the Mandalore isn't the only way to live, he's decided to leave the Mandalorians. By the end of the episode, he wasn't a true Mandalorian, as he appeared alongside the Armorer and Paz Vizsla. During Star Wars: The Clone Wars, the leader of the Death Watch was recognized as the Darksaber's true owner even though he wasn't a Mandalite.
One of the questions raised by Din's expulsion is why the armorer allows him to keep his helmet and armor, unlike the Darksaber, who does by right belong to the Mandalorians. The irony with it is that it came via him being exiled, rather than a choice, and that he had just proved himself to be the rightful owner of the Darksaber. The Mandalorian season 3 should be able to bring these threads together, with Din becoming a true Mandalorian and proving himself the rightful owner of the Darksaber, unifying what's left of the Mandalorians behind him. The Darksaber's Star Wars history is long and bloody but, as The Book of Boba Fett episode 5 shows, it can only be won in combat. While there would be a certain logic in the Darksaber only being used by an actual Mandalorian, and so one might've expected the Armorer to prevent him from keeping it, there's no firm rule to suggest that's actually the case. The Book of Boba Fett episode 5, "Return of the Mandalorian," delivered on its promise with the appearance of Din Djarin, but also flipped it. It didn't seem to be in question that the Darksaber belonged to Moff, even though Bo-Katan wanted to get revenge on him for his atrocities against Mandalore.
Even though he's no longer a Mandalorian after The Book of Boba Fett episode 5, he still has the right to wield the Darksaber. Even if he's not a Mandalorian, there's no doubt that he has won the blade fairly, and not even the armorer could deny him that. Bo-Katan wasn't the rightful owner of the Darksaber since she didn't win it in combat.
After being stripped of his standing as a Mandalorian, Din kept the Darksaber, raising more questions about the weapon's rules and place in the society. Din's claim to the Darksaber can't be revoked without him being defeated in combat.
There is precedent for a non-Mandalorian wielding the Darksaber.
The Book of Boba Fett episode 6 is available on Disney+. There is a warning for The Book of Boba Fett episode 5.