Just a cw answer since these are my own little conjectures. I don't quite have a good basis, but I have ideas based on a source outside the anime (eg the manga, the live action or an author interview).

Idea 1

My source is:

the 1st movie.

This source makes me think that the narrator actually has an identity, namely

Kaguya's cardiologist Dr. Shozo Tanuma. In the 1st movie, Tanuma and the narrator are portrayed by the same person.

And it's actually said explicitly

Idk. In this case, I guess the narrator calls or later starts calling Kaguya by 1st name due to knowing Kaguya personally but doesn't know anyone else personally and thus calls the rest either last name or full name.

Of course we see initially at some point in

the anime series the supposed narrator doesn't yet do this...

S02E08

and

Idea 2 (not mine)

Another guess I read elsewhere:

Also there's a reddit post that says Kaguya's dad is the narrator. I guess that would explain also things: Kaguya's dad calls Kaguya by 1st name and everyone else by last/full name.

Idea 3 (mine again):

Same flavour as the previous ideas: The narrator is simply someone who knows Kaguya personally but doesn't quite know the others personally or as well or at least calls them by last name but Kaguya by 1st name.

• Idea 3.1. But instead of (Idea 1)

Kaguya's cardiologist

it's like, say, a psychiatrist or psychologist that Kaguya sees in the future.

• Idea 3.2. Or could be anyone Kaguya confides in that doesn't really know the others or at least fits the bill of calling Kaguya by 1st name and the rest by last name. I think for example Nagisa Kashiwagi fits the bill. We see in S02E05 that Nagisa, in h own thoughts, refers to Kaguya as 'Kaguya-san' and Miyuki as 'Shirogane-kun'.

(However, in actual conversation, Nagisa might address Miyuki directly as 'Kaichou' or 'Shirogane-san' or whatever.)

Note 1: In S01E05, Nagisa addresses Kaguya as 'Kaguya-sama'. Well, at least it's not Shinomiya-sama, so I think this doesn't rule out Nagisa.

Wait, au contraire, perhaps this previous Kaguya-sama actually suggests Nagisa is the narrator because -sama is the honorific used in the title if none of the other named anime characters so far (besides Ai) call Kaguya as Kaguya-sama and if there isn't a result at the end of a story that includes Nagisa's name.

Note 2: Oh in S01E10, Nagisa says Shinomiya-san. Okay I don't know what to think anymore for Nagisa. Lol. --> https://anime.stackexchange.com/questions/66615/why-does-nagisa-call-kaguya-1st-kaguya-sama-then-later-shinomiya-san-then-later

Idea 4:

In the dub, Ai Hayasaka could be the narrator because AmaLee both voices Ai and dubs the openings.

Idea 5:

Chika is the narrator.

Akasaka said that he is like Chika in that he loves board games, and that he feels like her and has her personality when friends and family visit.