Does the Rennala fight in Elden Ring make any sense from a lore perspective?

I don't get why you're fighting her.

You fight her because she's a shardbearer. According to Gideon, you will need the Great Runes of at least 2 shardbearers in order to repair the Elden Ring and become Elden Lord.

It seems like Ranni supports you by giving you the spirit-calling bell early on

Actually, she's only fulfilling a request from Torrent's previous owner. At this point she neither supports nor opposes you.

then you kill her mother

That actually never happens. After the fight, Rennala will remain as an NPC that allows you to respec your character. Her arena becomes a non-combat zone, so killing her is impossible unless there is a bug that allows it.

then she takes you into her service afterwards as if nothing happens.

That's probably because nothing happened from her point of view. The moment you defeat Rennala, you can hear the following lines:

Upon my name as Ranni the Witch.
Mother's rich slumber shall not be disturbed by thee.
Foul trespasser.
Send word far and wide.
Of the last Queen of Caria, Rennala of the Full Moon.
And the majesty of the night she conjureth.

This happens even after you've met her and implies that she doesn't know who you are. And since she's nowhere to be seen at any point before, during, or after the fight, Rennala's second phase may be understood as either a spell that triggers automatically, or as a remote spell that doesn't allow her to know who you are.

In other words, even if she knows that something happened, she might not know you're the one who did it.