Even if you don't collect personal data, you need to comply with regulations to document properly the fact that you do not collect personal data.

how do you store chat inputs without collecting personal data?

That's possible in general but not for this application; a chat interface to an LLM isn't very useful unless you can tell it whatever you want—including GDPR personal data—and then pick up the thread of conversation later.

When regulations become sufficiently burueacrafied it's extremely easy to accidentally violate them doing completely normal things. As a really random example, in California when you operate a foodcart it's not enough to just keep your area and wares in sanitary condition, instead you need a dish washing bin of a minimum of exactly 'x' inches 10.5 IIRC.

A guy who was just ensuring he was preparing clean healthy food, keeping everything sanitary and all that might assume he was naturally obeying all regulations. But that assumption can cost one a big fat fine (leading to fun scenarios like a food cart vendor needing a compliance legal team), and given Musk's relationship with the EU - they'd love to crucify him him on any possible technicality they can find.