I always saw that sea-faring analogy more general. The spaceships and their crews in the 23rd century are a lot more like sailing ships and their crews in the 17th century. Planets are new shores, and space is the ocean between them. There's discovery, there's trading, there's dangers, there's myth and romance.
I guess that well devoloped interstellar traveling races would indeed create a proper coordinate system, so that ships would indeed travel in the same plane, relative to where they currently are. Every solar system has a specific plane, so if a ship is between two planets, it would probably move relative to that plane. If it's in orbit, the orbital plane or maybe the equatorial plane is the reference. If they are outside the solar system, the galactic plane will be the reference.
Agreed. I always felt the refit Enterprise in TMP
on the big screen of a movie theater was like a sailing ship of old Earth oceans.