Eirin gave Reisen the kanji (鈴仙) to write her name with, as well as a surname Udongein. That's how they made her name to appear more earth-like.
For the record, Eirin and Kaguya just call each other "Kaguya" and "Eirin".
Re: Kaguya's name: There's a brief scene in SSiB where the new Reisen talks to Eirin and Eirin's "name" is written as a bunch of unintelligible scribbles. The impression I got from that was that she has a "true name" that they're hiding from us for whatever reason. Anyway, this makes it seem fairly likely to me that Kaguya is also using an alias.
In the original text, it's written as 八意??. It... was an extremely clumsy reveal without proper setup.
Incidentally, you mentioned some time ago that DDC was set in autumn because ZUN put maple leaves in the charging animation. Were there any other reasons for this or is this the only hint as to what season DDC was set in?
DDC has plenty of seasonal symbols.
Falling maple leaves can be seen in several occasions during the story, from in-universe to "now loading" and music room.
The entire Stage 3 is a callback to IN; the full moon indicates that DDC, just as IN, took place during the harvest moon festival (the full moon of middle autumn).
Kagerou's dress is based on Hanafuda's "Pampas Grass and the Full Moon" card, an autumn card.
I'm now hijacking the topic to do an update of my "eight point star" symbol observation.
It's the same symbol that rotates on PC when you focus. It's also the background of ESoD difficulty selection screen. Another special appearance of the symbol is in IN. When a bosss charges up for a special attack in IN, the special effect is many "snowflake" symbols gathering on the character.
SaBND also referenced Yukari's symbol, when she was getting into a battle position to test the fairies' strength.
My personal theory is, this is the symbol of spell card rules.
In addition to the above, the symbol also appears:
In PCB, while the Stage 1-4 characters explode into snowflakes, their charge-up effects are not snowflakes, but this symbol, same as the IN characters.
(BTW, in IN, Mystia and Keine also share Wriggle's "blue blinking" effect, but only in their first boss spell cards.)
In PCB, when your bomb bullets hit an enemy, the symbol will "zoom in" on the enemy. This may indicate it's a crosshairs symbol. So when you focus, crosshairs appear on your hitbox?
In PoFV, the EX attacks are transported to the opponent's by this symbol.
In Fairy Wars, it's in the center of the difficulty selection screen.