Why is neutron star a star? I am wondering since neutron star is smaller than the average main-sequence stars and it doesn't produce energy via fusion so why do we still call it a star?
It seems Walter Baade and Fritz Zwicky may have coined the term in 1934 (see here) and the name stuck around.

I've known some astronomers to label them 'stellar remnants' for the reason you gave; they are not stars in the sense they do not produce energy via nuclear fusion, and that they are not made entirely of neutrons as the name would suggest.

They seem to be called stars for lack of another layman's term (from our point of view on Earth, not all neutron stars are pulsars), and that they still share some characteristics of stars: they emit their own light, they have masses greater than the Sun and can even have planets orbiting them.