Each tool remembers inside itself and nowhere else.
ChatGPT Projects keeps files and chats together, and does it well — inside ChatGPT. Claude has its own. Neither can see the other, so anything you settle in one is invisible to the next, and the usual workaround is pasting a summary into whichever you open.
That works for about a week. Then the summary is a fifth thing that might be current.
It is not the conversation. Most of a conversation is working out, and re-reading it is worse than useless to a model that will happily treat a discarded intermediate step as a conclusion.
What has to travel is the small set of things you concluded — what is current, what replaced what, and what you decided against.
Canonly holds the conclusions rather than the conversations, and any AI that speaks MCP can ask it what is current — so the answer is the same whichever one you opened, and it stays the same when you change your mind.