While illegible in both episodes, this starship model was labeled as the USS Melbourne (NCC-62043).  An Excelsior-class starship was also named USS Melbourne in the "Emissary". This ship is listed as unnamed, because Memory Alpha considers the name of this ship has been retconned. See USS Melbourne for more information about the model and the naming issue.
The footage of this vessel in "Unification I" was a reuse of the footage from the Wolf 359 debris field from "The Best of Both Worlds, Part II", though the damaged sections of the hull were clearly on fire in the "Best of Both Worlds, Part II" version.
Initially the ship was disabled as it passed through a cloud of tetryon plasma, but later caught up with the Yellowstone and subsequently damaged the runabout's warp drive, causing the runabout's destruction. (VOY: "Non Sequitur")
Although not identified visually, it is possible that this ship is the otherwise-mentioned USS Lexington, which was referenced in comm chatter during the engagement.
On the other hand, it is concurrently possible that this ship was the USS Endeavour (NCC-71805), as per the Star Trek Encyclopedia (3rd ed, p. 135). Some prints of Star Trek: First Contact do not include shots of this vessel.
This appearance marked the last time that use was made of the physical studio model as a production asset. With the exception of the USS Enterprise-E, all other Federation starships in the sequence where executed as CGI models. For further information on the model, see Nebula-class model.