Magnet wire is a generic term for wire intended for winding coils, with very
thin insulation so as not to waste precious winding space.

Enameled wire is simply one form of it, probably the most popular. Enamel
is a lot thinner insulation than a plastic sheath so won't waste winding
space.

In the old days there were enameled, single cotton covered (SCC), double
cotton covered (DCC), single silk covered, and some others. Today enamel is
used as well as various plastics. One popular type melts away with a
soldering iron, making stripping the end unnecessary, very convenient.

They are all magnet wire.