I was in it yesterday. It’s part of the tube network. It’s on the tube map AND it’s fully interchangeable with the other tube lines, AND the fare structure is the same as the rest AND the announcements are the same AND locals call it the tube, they regard it as the tube AND, importantly, in central London it runs through a tunnel, hence the nickname tube. Note the ANDs there.
Think of the definition of a duck. If it looks, works, costs, operates, is named and even smells like the (traditional) tube, it is.
Might an analogy be the Sydney motorway network. There is a series of highways named M1 M2, M3 etc. Some are government owned and free some are privately owned and toll roads. They are fully interchangeable with each other. They all have the same car and truck capacity as each other, the lines are all marked the same and they appear on various maps illustrated the same.
Are the privately owned M roads not part of the “Sydney highway/freeway network” because of their ownership structure?