So, taking a some random offshore tug/supplier (which had twin screw propulsion, as I understand), cutting off everything above the main deck, cutting off her bow part, getting rid of whatever is inside her hull, longitudinally cutting the hull in halves, adding longitudinal midsection & welding the port & starboard halves to it, adding new engine room (single screw propulsion) & other equipment, adding new stern part, adding new bow, building the hull above, adding superstructure & more equipment... Is it still the same ship or it's a brand new ship?

Unbelievable but apparently true. I do not think the Food and Agricultural Organization of the United Nations would have made a mistake http://www.fao.org/figis/vrmf/finder/!/display/vessel/IMO/7390416
Also from original ship builders site - Mangone Shipbuilding USA ..launch number 114 Sea Bruse of Bugge Supply Ships of Norway, Anchor Handling Tug/support vessel. Later Bruse, Alexandre, fishing vessel Saga Sea, Kamchatka, Saga Sea (Norway)

Then it's some unbelievable conversion! On http://www.shipspotting.com/gallery/photo.php?lid=1688228 the flat/angular underwater hull part would the the only visible proof of her originally being a classic offshore supplier.

Correct IMO according to Equasis. Photo shows as tug (499 GT) of Fred Olsen before conversion to fish factory ship in 1990 (4848 GT).Increased in length by approx 40m and beam by 5 m during conversion. Latest name Saga Sea since Kamchatka.

Wrong IMO?