How can I safely identify a single asset-folder?
Moving a folder is a pretty complex operation. You have to move all the files contained in the folder, all of the subfolders and all of the files contained in the subfolders. Another thing to consider is that a move is essentially a copy+delete.

So, how Assets goes about all of this is that firstly it mirrors the subfolder structure to be moved in the target destination. It then moves the files one by one. After that, it deletes all of the old folders.

Dividing the operation into many steps accomplishes several things:
1) It allows for displaying the progress bar.
2) It divides the processing time and memory needed across multiple requests, helping prevent timeouts
3) If the user closes the browser window, there is at least fault tolerance and no files are lost - you just get duplicates of some files and you can retry the same operation using "replace" instead of trying to figure out which files were moved and which were not.

So, the folder ID changes, because it's not the same folder anymore. Theoretically, it would be possible to manually adjust the folder IDs afterwards so that they remain unchanged, but I don't think that's a good precedent.

I could argue that folder IDs are a Craft internal thing and relying on them is not encouraged, however, I would just offer a better solution UX wise - just use an Asset source type drop down selector and a text box for the relative path. Not only it would be less amibigious than a magical ID, it would also be more consistent with user experience of addressing folders while also pointing to a specific spot, instead of an entity, which is definitely better for long-term maintenance.