Not having used anything except for Firefox, I don't have any experience with migrating to different browsers. However, my understanding is that Chrome shows a little pop-up that lets you import from previous browsers rather than relying on the user to do a data export. Correct me if I'm wrong about this.
I suspect that Claude couldn't make an "import from ChatGPT" button because OpenAI would make it difficult, so they'd have to rely on user initiative and technical capability (exporting to JSON and importing from JSON is enough technical friction that the average user won't bother).
Try telling your PM that you want to ignore Safari when you create your website with 60%+ of mobile users in the US using iPhones and globally your most affluent users are on iPhones. Even if they download Chrome for iOS, they are still using WebKit.
You are vastly overestimating people's willingness to deal with bullshit, when the product does not have a real lock in.
It would be incredibly easy to have a company offering their ChatGPT over WhatsApp or iMessage, and get people to start using it instead of an ad-ridden GPT app.