Grouping of the connections

I use HeidiSQL very intensively and currently have 34 connections in the connection manager.

Here I would like to have a grouping function, for example, group by projects. It's quite confusing when there are more connections.

Well you can create folders, subfolders and so on, and move your sessions into them:

To create a folder, click the down-arrow on the "New" button, then click "Folder in root folder". Moving a session can be done via drag'n drop.

oh ok, i mean in the dropdown^^

What is "the dropdown"?

In a similar vein... what I would love to see is a session search filter above the session panel which filters on input all displayed sessions (folders and connections, when a matching entry is "found", all parent folders would need to be shown as well)

Nice idea. You should report that filter as a new issue in the tracker.