I believe we can do that.
You open the Artboards panel and you re-arrange them :)
But I suggest you mean that re-arranging items in the panel should make artboards actually rearranged on the canvas, right? To mimic the behavior of pages in InDesign?
This is indeed an interesting idea, but the problem is that artboards can be of any size and positioning. We can have artboards inside other artboards, they can cross each other partially. What should Illustrator do then? Artboards are not pages, but more like viewports, although they can be used like pages.
So if you use them as pages, you can change the order of artboards in the panel and then do Object > Artboards > Rearrange All Artboards, and it will put them in the correct order you defined. It’s some what limited and forces to change position for All instead of any local changes, but it’s there already.

Please tell more about the workflow that requires the feature.

I wouldn't consider this as a bug, but I would really like to figure out how to get this functionality. The only way I can see what artboard is associated with the name is to go into the Artboard mode, or in the export panel, but in both ways, I can't copy elements from the canvas and paste as names on the artboards. I often add labels to be able to search a specific artboard, which means I want to copy and paste from the selected entity to the artboard name, but I need to keep switching modes to do that, which is very tedious, especially if I need to do this operation multiple times.
You have this capability for Layers, but not artboards. Why??

It would be great to be able to drag and re-order artboards when designing multipage documents in illustrator.

It would be great to be able to drag and re-order artboards when designing multipage documents in illustrator.

I am always working whit a lot of artboards. The way how now is working it is tedious to work because you see just the name of the artboard. It would be great for changing the order of the artboards to drag and drop the thumbnail of the artboard the same as in Adobe Acrobat Pro - Page Thumbnails view.

That would be a great feature!
Ideal when working on many artboards and not having to go to the re-arrange panel everytime.