OK, I tried again with the same results. Here are the steps I took.
Whilst your screenshots don't indicate which browser you are using, the methods to solve this issue should be the same across them all. Most of this information will come from my own understanding, and some of the more technical aspects may be incorrect or wrongly worded.
As mentioned before, Save files in most HTML games on F95 is handled through the same inbuilt system, wherein the save data is stored in your browser's cache.
In Incognito Mode, your browser doesn't store any new data in the cache for later use after that session. And saves made in incognito mode will be deleted once the browser is closed. Alternatively, most browsers have an option in the settings to "clear browser cache on exit". This option basically does the same thing as incognito mode, deleting any stored save data in the cache when the browser is closed.
There are three potential solutions to this issue, and these solutions will work across the majority of HTML games on F95:
1) Avoid using Incognito Mode
The simplest option, but also the one people tend to avoid due to the type of content being access - People don't usually want it in their browser history.
2) Uncheck the option to "clear browser cached on exit"
Googling that phrase with your browser's name should indicate where to find the setting. Turn it off and your browser's cache and the game's save data will be stored for future use. This method will not work if you are playing in incognito mode.
3) If you continue with Incognito Mode, you can see the option in the save menu to "save to disk".
This creates a downloadable '.save' file that can be stored offline. The file size is minimal, can be renamed without issue, and can be kept anywhere on your PC. Clicking "load from disk" allows you to open that '.save' file in the game to pick up where you left off.
These are the main solutions that I can see, and as mentioned this will work for the majority of HTML games on F95 Zone as most HTML games here are made using the same engine and code format. Your particular issue is more to do with your browser mode/settings, and less with any of the individual games.
Let me know if these solutions work out for you, or if the problem still persits.