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.
According to wikipedia:
A star is a luminous sphere of plasma held together by its own gravity
then it goes on to say that
Near the end of its life, a star can also contain degenerate matter
(see also nuetron degeneracy).
- Any of the heavenly bodies, except the moon, appearing as fixed luminous points in the sky at night.
- Astronomy. any of the large, self-luminous, heavenly bodies, as the sun, Polaris, etc.".
Nothing in those definitions explicitly requires fusion, though the mention of plasma seems to infer it. The quantum mechanical effects in a nuetron star are an analogue to an ideal gas in classical physics. The degeneracy pressure thereof prevents the star from collapsing into a black hole.