The classification of stars using spectral class is a very useful classification when considering the properties of (the atmosphere of) a star at that moment. If you consider the different stages in human development (embryo, fetus, infant, toddler, etc...), for instance, here one person also continuously changes its class.
So it is not strange/uncommon/forbidden to change a classification of a thing whether it is a human or a star.
It also does not mean that other classifications are not useful. It depends on how you want to use the classification. In literature, concepts such as low-mass star, or Sun-like star are also used.
Classification according to the evolutionary stage (i.e. the type of fusion going on) of a star is also very common.
See the image below from ATNF in which you can see how stars of different zero-age main-sequence mass move along a colour-magnitude diagram. If we follow the track of a star of one solar mass then we get the following subsequent evolutionary stages:
- MS - Main sequence stars (spectral type V) fuse hydrogen into helium in the core. When the hydrogen in the core is depleted, the star becomes an RGB star.
- RGB - Red Giant Branch stars (V->II/III) fuse hydrogen into helium in a shell around the core while the core is inert helium. At some point (Helium flash) the core temperature becomes high enough to start fusing helium into carbon and oxygen, the star then becomes an HB star.
- HB - Horizontal Branch stars (II/III) fuse helium into carbon/oxygen in the core. A shell with inert helium surrounds the core, which in turn is surrounded by a hydrogen burning shell.
- AGB - Asymptotic Gian Branch stars (I) have a carbon/oxygen core with several inert or fusion shells around it.
These types (MS/RGB/HB/AGB) are classes in a very useful and common classification scheme.
In the end it just depends on what you want to do with your classification scheme.
NB. The classification of the Sun changes from G2 to M because the outer envelope and atmosphere expand enormously, causing the atmosphere to cool down. A cooler atmosphere means that the Sun changes from a yellow (6000K) to a red (4000K) star. The wavelength, and therefore colour, at which most energy is transmitted depends on the temperature of the star's atmosphere.