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
here
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").
Best Answer
More generally, using an article before a proper noun that doesn't have one built into it (as the United States and the Rolling Stones do) is one example of using a proper noun as a countable noun.
There are several reasons why we might do that normally. One is to say something like "there are three Johns in the group", meaning "there are three people called John in the group".
Another is to add distance to the identification; "I have a John Smith on the line" is a common expression for "I have someone on the line, who tells me he is John Smith, and that is all that is known about him". A similar is to report, e.g. "One John Smith is accused of the crime", emphasising that we have no further identifying details at present, and hence we are not stating precisely which person of that name is the subject of the sentence.
Another is to use a proper noun as an example of particular traits that could also be held by others (a type of synecdoche). "The next Bob Dylan" (a singer-songwriter from the folk scene who will repeat Dylan's success), "He's an Einstein" (he's very smart), "All Mozarts have their Salieris" (not really true even for Mozart and Salieri, but let's say we believed the film Amadeus was accurate).
Another, almost inverse to this, is to speak of the person or thing signified by the proper noun at a particular time, or from a particular perspective: "The London of a hundred years ago was a notoriously unhealthy place", "The John you know is not the John I know" (that could also mean you are talking of a literally different person, depending on context).
The above are reasonably standard, though figurative.
Another common variation is to jokingly make use of these forms, when one normally would not. If talking of a friend, we would generally use their name as a proper noun, because that's how names work in English, but since every person called George is "a George", and so on the form is logically correct, though not strictly good English. To use it of a friend could suggest that you have gotten as far as knowing it's a George, but not which one, or that George's are all alike and you've hence found someone who will have all the George-like qualities that George has. Both obviously are not sensible, but therein is the joke. Another variant would be if you were looking for George, and then spotted him. Again "ah, there's a George" would suggest that you'd were just looking for Georges generally, which again is not sensible, hence the joke.
All of these last cases are examples of deliberately bad English, used as a joke, rather than something that would normally be considered correct.
[A completely different case is when there's a word that is the same as a proper noun, but isn't a proper noun, of which some slang cases started as a proper noun and are hence sometimes capitalised.]