I believe "Tige" is indeed a shortening of Tiger, and would be pronounced like tide with a hard g in place of the d.
From a story in the Atlantic Monthly published in 1860, apparently by Oliver Wendell Holmes, Sr (father of the famous American jurist by the same name):
Tiger, or more briefly, Tige, the property of Abner Briggs,
Junior, belonged to a species not distinctly named in scientific
books, but well known to our country-folks under the name "Yallah
dog." They do not use this expression as they would say black dog or
white dog, but with almost as definite a meaning as when they speak of a terrier or a spaniel. A "yallah dog" is a large canine brute, of
a dingy old-flannel color, of no particular breed except his own, who
hangs round a tavern or a butcher's shop, or trots alongside of a
team, looking as if he were disgusted with the world, and the world
with him. Our inland population, while they tolerate him speak of him
with contempt. . . . Tige was an ill-conditioned brute by nature, and
art had not improved him by cropping his ears and tail and investing
him with a spiked collar.
Apparently, a yellow/yaller dog was a term for the breed now known as a Carolina dog (see Wiktionary and also this well-documented discussion).
However, I suspect that Tiger nn Tige was probably a name given to any yellowish or orange-y or otherwise tiger-like dogs (for example, if they were particularly large or ferocious). Buster Brown's Tige was an orange-brown; he is apparently commonly believed to have been a pit-bull, and was portrayed that way in film shorts (you can see the shorts in this blog post).
Looking at Ngram, Tiger/Tige seems to have come into vogue as a dog name sometime around the time that Holmes wrote his story, but I don't believe he originated it.
There is an 1837 story by Edgar Allen Poe with a dog named Tiger (in that case a Newfoundland, which are large but not yellow), an 1847 story that mentions a "little yellow dog, Tiger," and the earliest example I can actually see any part of is from 1791 (combining snippets):
...Mr. Stinton's great dog Tiger made his unexpected appearance
upon the stage, and joined in the fighting scene betwixt Posthumus and
Jachimo. —E. Cave, from The Gentleman's Magazine, here and
which suggests the ferociousness aspect.
Interestingly, an 1857 novel by Francis Butler entitled The Life and Adventures of the Celebrated Dog Tiger features a bull-dog/fox terrier mix (who himself narrates the book) who is apparently known for NOT being yellow, deplores the quarrelsome reputation of bull-dogs, and leaves it to the reader to determine "[h]ow far [he] may have merited this ferocious title".
Some of these sources might have been influential, but they look as if they are following a pet-naming trend, rather than setting one (particularly those that appear to be relating stories from real life), so it's hard to say whether there was a particular "source" for this name or whether it spontaneously occurred to various individuals as a good dog name.
For a more recent example of a dog named Tiger, see the Brady Bunch dog, who was tawny and largish.
**Edited to clarify the first sentence.
**Edited again to add, as requested:
As suggested by @niallhaslam, Tige used for contemporary dogs may be an Anglicization for the Gaelic given name Tadhg. This was the name of a medieval Irish king. See Behind the Name for more details. Its diminutive form, Anglicized as Teagan, might be more familiar to (American) English speakers.
However, it is unlikely that this derivation accounts for the bulk of nineteenth- and early twentieth-century pets called Tige, because giving animals "people" names is a relatively new innovation. This article by Stanley Brandes offers fairly compelling evidence for this anecdotally-observed phenomenon (since pet names aren't centrally registered the way human names are).
On further investigation, one possibility that would make the Tadhg theory somewhat more likely, at least for some dogs Tige: the Anglicization "Taig" has apparently been used as both a term of ethnic pride by Irish Catholics and a racial slur by British Protestants referring to those same Irish Catholics (see Wikipedia's entry on Tadhg).
Dogs who are looked down upon (e.g. "cur dogs"—thanks for the phrase, @Sven Yargs) are often evoked in slurs (in fact, the poem cited in the Wikipedia article on Tadgh explicitly contrasts "Cromwellian dog" with the "Popish rogue" who eventually identifies himself as "Taig"), so it would make sense that naming of those dogs could flow the other way. Thus this connotation of the sound "tige" (hard g) might have influenced some instances of dog naming. In that case, I would expect Tige to be most popular as a dog name in areas with a significant anti-Irish Catholic sentiment/population, and significantly less common as a name for dogs owned by Irish Catholics in that era. Unfortunately, the data available to me is not detailed enough to properly draw or refute this hypothesis.
Related question: When did we start naming our dogs Rover, and Why? (which handily also addresses "Fido").