Classical economics is something different -- think Smith and Ricardo. Marginalism is sometimes called neoclassical, but is distinct from classical economics. The neoclassical synthesis was macroeconomics from the 1940s until the stagflation era. Micro-founded macro was later. I don't think neoclassical has a very clear meaning -- I would call any kind of Keynesianism non-neoclassical, personally.

Anecdotally, it's because so many people are bullshitting when they claim that something is a foundational hypothesis of the mainstream. People prefer to argue against a caricature rather than the real thing.