Of course, Jeph's narrative purpose for Hannelore getting Tilly's name wrong is unknowable, but I feel motivated enough to take a proper look at this narrative arc and see what meaning we can wrestle from it. Just bear in mind that this is my personal reading only, and can't claim it to be universal.
Let's go.
Number 3607: A Noble Steed'Ah, yes. Taffy, or something like that?'
''Thank you, mother," you'll say. "You're always correct, ..."There's nothing like the image of animal abuse (robo or otherwise) to quickly set up the villain of the piece. But this is also a thematic setup. Beatrice is disregarding the agency of the horse, subjugating it to her will. Maybe she'd even argue that this is best for the horse. It's easy to see an analogy between this and the way she is treating Hannelore. So this sets up the basic conflict of the arc.
Number 3609: Unstoppable'You are exactly the kind of person my mother would hire.'Tilly is exhibiting the same trait here - Beatrice's behaviour has rubbed off on them. So now we can see the basic conflict play out between the two of them.
Number 3610: Use Case'I just don't think there's anything for you to do, Taffy.Hannelore gets Tilly's name wrong for the first time. Remember Beatrice's claim that she is "always correct?" In spite of Hannelore's fury at her mother, she is, on some level, still accepting what Beatrice is saying. This will become clearer later, but this is the first hint that both Hannelore and Tilly are being fundamentally misled by Beatrice. Yet, the conflict will need to play out between them before the root cause (Beatrice's disregard for others) can be properly addressed.
Number 3612: Daily Abultions'Oh, I'm going to enjoy working for you, ma'am!'The first hint of the potential for them to become fast friends if they can resolve this conflict.
Number 3614: Dogs ARE Good'How long have you been working for my mother?'And the first hint of Hannelore's realisation that Tilly's behaviours have been strongly influenced by Beatrice.
Number 3617: Laying Down The Law'Isn't that pretty messed up, though? To disregard someone else's agency like that?'Hannelore finally addresses head on the central conflict between her and Tilly (and with Beatrice, by extension), and in doing so, breaks the spell Beatrice has over Tilly.
Number 3624: She Really Blue ItHannelore realises that she was essentially guilty of the same thing - hurting someone after being misled by Beatrice.
Number 3635: The Other Shoe Drops'We're going to see my mother tomorrow.'Title just about says it all. Hannelore realises the truth about her mother.
Number 3638: Just So We're Clear'I thought you were being your usual out-of-touch self ... But this wasn't about helping me at all.Climax of the story. Speaks for itself.
Number 3640: It Will Come Out In The Wash(very meta title)
'I'm sorry, Tilly. My mother used us both.'Apotheosis. What has come out in the wash? Hannelore and Tilly are now fast friends after realising that Tilly's behaviour and Hannelore's misnaming of Tilly had the same root cause: Beatrice. That's not to absolve either of blame for what they did, but to recognise that what they have in common is far stronger than their past differences.
To cut a long story short (
too late): Hannelore's misnaming of Tilly feeds into the story of how both of them were misled into thinking that Beatrice was looking out for them. It gave Hannelore empathy with Tilly's plight, as Hannelore had hurt Tilly after accepting something said by Beatrice, just as Tilly had done. It also revealed a quite positive aspect of Tilly's personality - they were quite patient in a situation where many others would have been aggrieved.
Hannelore's realisation of Tilly's real name is also kind of a trigger for her realisation that Tilly is, in spite of their mistake, a good person, and that Beatrice is the one who truly deserves her wrath. You could even see it as a metaphor for Hannelore's inability to see the real Tilly - she was looking at her mother.
I think that's the most effort I've ever put into analysing a QC arc, but I think it was one of the more interesting ones.
I do hope that forumites who have never forgiven Tilly's initial behaviour can finally come around to reevaluating them, just as Hannelore has managed to do.