Applications can define “tags” to store arbitrary metadata (such as band names, units, scaling factors, etc), but GeoTIFF does not have a strong metadata standard, so you end up with metadata in tags that are only useful with the application that created the GeoTIFF file. NetCDF4-CF has a useful metadata standard, and is becoming widely supported by current GIS software, so you may want to review whether you really need GeoTIFF format files. If you are supplying data to GIS users, you may need to encourage them to try NetCDF4-CF in their GIS system – in my experience, GIS users quickly learn to love NetCDF4-CF for the much more complete metadata that saves time and effort to reproduce when using GeoTIFF files. If you are stuck with GeoTIFF you need to look for application-specific TAGS.
You can try Python script using Band.SetDescription.