What happened to the project support?

It was removed from Kate 2.5 (shipping with KDE 3.5) because it was working so poorly for the most users and caused some confusion. Instead Kate 2.5 and up supports named sessions.

You can read more about named sessions here

Since a while, a very good replacement for people loving the old project support is available in form of the “Kate Project Manager Plugin”. It can be found here on www.kde-apps.org.