He's Mr Grinch, so it's at least his last name (and also explains him being THE Grinch who stole christmas). It's possible that it's also his job (see "Smith"), but
prefer that grinching is named after him, a la Scrooge
I thought it was a character trait. I assumed he had a "Who" name, but no one uses it because he was being a dick all the time so they called him "the Grinch."
I can't find a specific image of it, only a reddit post talking about it, but in Grinch (2018) he buys green hair dye. I interpret that to mean he is just a normal whoville resident, but dyed green. I suppose his green grinch hair could be graying and he wants to look young, or he could just be some other color and chooses to be green, but I like to imagine him as some crazy naked hermit that makes himself green and everyone else just puts up with it.
The entire song is derogatory. If you sing, "You're a butthole, Mr. Stinky", then you haven't identified who you're singing about, unless someone is actually named Mr. Stinky.
It's most likely either a species (like the sneetches) or, to my mind, a descriptor given to him by the narrator. Keep in mind no character (namely the grinch, who does speak) ever refers to himself as such. Sort of like if someone narrated your life and called you "the grouch".