I edited the message to make it a little clearer.
But to answer your question, no the mod is always required, if you so choose to use it. The message you quoted simply mean, don't use an old mod when a new game version comes out. e.g. if you look at the mod post you will see the specified game version:
Mod Version: 1.16
Game Version: 0.89
That means, mod 1.16 is for game version v0.89 and only for that version. When v0.90 comes out, you have to wait till the mod gets updated to use the mod. If anyone use the old mod on the new v0.90, they will certainly gets bugs that would be their fault. As this is also made known to the user, when using the mod for the first time. The mod tells you in-game specifically which game version it is for. I hope this is clearer to you now
. The current mod is working fine with the latest version of the game, which is v0.89
Go into your old game version folder > www folder. You would see your save folder.
Copy this save folder to your new game folder at the same directory > www.
It should look something like this: MNLv0.89/www/save
Once you verify your saves are loading up on new version, you can delete the old game version to save space.