The platform site’s API call playground history is not stored on the server. It is local storage on the browser.
It is not “cookies” being cleared, but the browser function that you use to “clear cache” (or what have you) that may be affecting more than cookies.
> Indexed DB: \platform.openai.com → keyval-store → keyval → oai:playground:chat:history
The odd thing: the playground site scripts have no need to enforce a 30-day history where they clean up your own local storage. Where you revisit 30 days later and everything that was there, being happy, gets wiped. A maximum number of turns would be a better management policy.
The subscription to ChatGPT doesn’t affect anything API.
If you want a robust session history - write an app!