It isn't class warfare if the two (or more) sides aren't divided by class.
What OP meant was the income and wealth gaps are they only things they care about, not the only things that are real.
No, it's the empirical fact that the US is a plutocratic oligarchy. The oligarchy runs the economy, the government, and the media for its own benefit, and is violently opposed to anything that challenges this.
https://www.cambridge.org/core/journals/perspectives-on-poli...
Taxonomy is never perfect. What's the most obvious one that seems watertight? I'll tell you the holes in it.