Wood between the windows is rotting, how to fix this

Looking at your pic and arrows, it looks like the bottom sash does not fit well. If all the factory weather seals are in place, not damaged or worn, I think they may be poor quality. That is rare since most windows in the last 10 years or so are normally fairly tight. The area that really concerns me are the sides where you show the arrow. This area usually depends on a snug mechanical fit, not weather stripping to seal the unit. Does this lower unit seem loose? Leaking at the joint of the two sashes is common and can be fixed with new seals most of the time, but the sides do not. Also, some windows need to be locked for the sash to sash seal to fully engage.

Caulking is not a good way to seal these kind of leaks if you ever want to open them again!

Maybe you can more fully describe the areas in question and the fit of the windows.

I had never seen these before, but I was finally able to track down some information and replace them. My windows bear a "Viking" name, but I didn't find anyone selling parts under that name.

However, the useful term for these are "channel balance" or "block and tackle balance" windows. With that term, it's possible to find replacement parts and even some youtube videos showing the process of replacement.