I would like to have the functionality shown in the following video on Inkscape (preferrably) or GIMP: https://youtu.be/rsFaQZAIZZI?t=18s
The user should be able to draw a rough circle, square, rectangle, triangle etc. with a pencil tool and the application should detect its shape and turn it into a crisp shape (vectorized, if possible).
Does this feature exist in either application, either built-in or as a plug-in?
No, Inkscape does not have a tool with the ability to detect the shape of a free-drawn shape and convert it to a vector version of that shape.
There is however, a Wishlist item in the Inkscape bug tracker from 2011 asking for this feature: https://bugs.launchpad.net/inkscape/+bug/714745