Bobtail

This is highly dubious at best. The lyric (verse 1, line 5) is "Bells on bob tail ring". This is rendered exactly the same in every version I've found online, including both versions ("Lyrics" and "Original Lyrics") on Wikipedia. Note that "bob tail" is written as two words, with no initial capitals, suggesting that it's not the name of a horse or any other creature (human or otherwise).

In answer to a query on WordReference.com, someone wrote: "Horses often have their tails 'bobbed'. This generally just means that the horse's tail is folded up and tied (like an old lady's bun – sort of) to make it look shorter and get it out of the way. If you put bells on a horse's bobbed tail, they will ring as the horse moves."

Someone else on WordReference pointed out that there's also a "bobtailed nag" in Stephen Foster's Camptown Races.

The final verse of Jingle Bells includes the following lines: "Just get a bobtailed bay, Two forty as his speed".

It seems clear to me that the Jingle Bells lyric only says that the horse has a bobbed tail. Its name is not mentioned.