Does OpenRouter eat this cost to get their hands on a copy of the conversations people are using with the model?
No, this is OpenAI doing the discount, not Openrouter by themselves. OpenAI is crushing it with their 5.6 models, and they probably decided there was no better time to grab as much market share as possible.
OpenRouter offers 1% discount to save your conversations, explicitly opt-in.
Anything else they don't save it. Even if they tell you the model provider saves your data for training.