Imagine the actual dread if the next Sky Clan deputy is like, Rileypool, or Harveymoon (yeah I know some of those did not make it to the latest books, but you get the idea) and then the leader dies and Star Clan is like ''Harveystar? Rileystar? Sorry it's just so stupid your name doesn't mean anything."
Like, there's an actual reason why loners/rogues/kittypets joining a clan get a new name (Rusty -> Firepaw, Red -> Russetfur, Leaf -> Leafdapple, etc) and that reason is that your whole identity circles around your name and if that name is not part of the naming system in place then you feel (yourself and in the eyes of others, including the reader) out of place.
But also the naming system is one of the few things that identify the Clans as something noble and different, some ideal to attain... it's what makes them different, at first glance, from Blood Clan or Djinn's group. It's the symbol of the warrior codes, the taking care of the elders, queens and kits, the communion with Star Clan...
And for a clan, who is xenophobic in building, accepting an outsider, giving them an actual clan name, making them part of their family as if they had been born in it... it's the highest honor.
So it infuriates me when Millie or Boulder are like ''yeah actually I prefer my name which was given to me by twolegs or loners rather than the name my actual (adoptive/found) family wants to call me", it's like, you left your older life behind you for a reason, and that reason was that you prefered to join a clan. In most of those cases it's not even cats who are like ''yeah but my family or my old friends might be confused/might look for me and not find me'', because the clans are all that's left to them.
It's like you're an orphan who could only left your abusive/neglecting family after they died. You're taken in by a friend's family. You don't have much good memories of the times before you arrived in the family. You convert to the family's faith. You share all their political, religious beliefs. You moved to their country with them. You adjusted to a completely different way of life. You feel happier and more in your place in here than ever before. You want to leave your old life behind you. And you're in a society that allows you to change your name as long as the head of the family accepts it. And so the head of the family is like ''hey kid, from this day on you can take our family name" and you're like ''oh wait no, I'd rather retain the name my neglecting family gave me, even tho I despised every single one of them, because it would feel like you're trying to take over my identity by giving me a new name". And the head of family is like ''well not to be mean but you did change your identity by becoming one of us. You changed your beliefs, your opinions, you told my husband that he was like a father to you and my kids that they were like siblings to you. You told me that you wished your mother had been more like me because then you would not have needed to run away and risk starvation in the streets. You told us all you wanted to put this life behind." and you're like ''yeah but still, I'd rather keep the neglecting family name."
So Sky Clan found a way to mix it and said ''ok you can keep your family name, and you can hyphen it with ours if you want." and the kids were like ''yeah great idea" and that's how you get Harveymoon and Rileypool.
The other clans however, are like ''no, you can stay in our family but if you want to keep your old name you cannot have ours too, because we despised your old family for what they did to you and we don't want it to be hyphened with ours". That's how you get Boulder, Millie, Daisy, Purdy... in the middle of Graystripes, Blackfoots, Clawfaces, Whiteclaws, Lionhearts and Molewhiskers.
It also infuriates me on that point that Firestar was like ''hey Millie it's been two moons (or idr the actual time) since you joined us, it's time you get a clan name" while with Daisy in the background, who had been there for like ten moons and taken care of numerous kits and queens, protecting them during a freak badger attack etc, he was like ''yeah no you can keep your loner name i dont care".
Oh, ok, so Graystripe's girlfriend is accepted right away while the one who actually has an important, unique role in the clan must still prove herself? Or is it that Daisy nearly never leaves the camp while Millie was seen by everyone at the Gathering she arrived, and you want to make sure the one the other clans know about is made a full warrior to cut rumours short? Well in that case you should either have named Daisy Daisyflower or something and everyone kept calling her Daisy, so that all cats in the clan had warrior names, or you should just not have given a choice to Millie (and also discuss it with her in private before putting on a show and being publicly ridiculed when she refuses her new name).