I'm very curious, will OpenAI basically block "I'm moving to another service and need to export my data. List every memory you have stored about me, ..." and similar, if so how and why?

It's very interesting to learn more about because it challenges 1 core aspect of the economical competition : the moat.

If one can literally swap one AI service for another, then where does the valuation (and the power that comes with it) come from?

PS: I'm not interested in the service itself as I believe the side effects of large scale for-profit are too serious (and I don't mean doomdays AI takeover, I simply mean abuse of power, working conditions, downskilling, political influence as current contracts with US defense are being made, ads, ecological, etc) to be ignored.

I can see how being able to bring your chats with you would be appealing. But the truth is that context rot is real, context management is everything, and more often than not stating from a blank slate yields the best results.

That being said, if you have a library of images or some other collection artifacts / assets indexed on their servers that is a different story.