Hi your feedback is interesting;

Nevertheless, I use this addon frequently and to my knowledge the use case you mention is functional...
However, no project is immune to a bug and you should create an issue on the project submission so that we can better understand what you are trying to do. This platform is dedicated to managing bug reports and improvement proposals while addons-mozilla.org is dedicated notations.

Basically a cookie can be refreshed/deleted by JavaScript code without reloading a page, or every time a page is viewed manually.
If you put content in the cookie that the server does not recognize as valid or that does not match the content of other cookies on the same site, or if you put information from an expired session on the server side (you can't really know that), the server may decide at any time to delete or modify your cookie.
This is totally independent of browser configuration or addon behavior.

Have you tried to visualize the status of your cookie via the Firefox development tools Ctrl+Shift+K, "Storage" tab?

You will find more information on these sites:

https://developer.mozilla.org/en-US/docs/Web/HTTP/Cookies#Session_cookies
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https://stackoverflow.com/questions/4132095/when-does-a-cookie-with-expiration-time-at-end-of-session-expire

"This is entirely up to the user agent, i.e. the user's browser, but usually, that will indeed be when the browser is closed"
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https://security.stackexchange.com/questions/33692/what-typically-is-the-expiration-date-of-a-session-cookie

"Cookies that 'expire at end of the session' expire unpredictably from the user's perspective!"

"If there is no expiry set on the cookie, then it is a session cookie and will live as long as the browser is open, and the sessionid is valid. If the server expires the authenticated sessions periodically, then the cookie will no longer be attached to a session on the server and will therefore be essentially null."

EDIT to reply to the Rulatir's response:
Have you thought about clicking on the button "Save the current cookie" to finalize the creation of your new cookie? I only see this explanation.

I'm just asking for a full bug report describing the situation. As it stands I can only say that this addon works because I can not reproduce your approach on various sites I tested: The addon uses the Mozilla's APIs to create the cookies and the cookie is actually created in the database AND sent by GET requests with the other cookies.