I suspect that "Old Toby" is mainly present in order to supply a fictional etymology for the word "tobacco".
We know that Tolkien dedicated effort to multilingual explanations for things like the Brandywine River being a Westron pun ("Bralda-him" versus Sindarin "baran-duin"), golf coming from the name of the orc Golfimbul, and so forth. Some of what he does is "serious play", like Eowyn's alias Dernhelm deriving from Old English dyrne, a secret; she is literally hiding under a helmet. Much of it is more lighthearted. The internal etymology and herb-lore of tobacco is remarkably extensive, which if nothing else shows that Tolkien was personally engaged by the linguistic and world-building games around it.
Overall, I think it's fair to conclude from the evidence presented below that:
- Tolkien thought a lot about fictional etymologies,
- was bothered by words like "tobacco",
- had already put the word in Middle-Earth due to The Hobbit,
- wanted to include a tob- element in the name of the first hobbit to cultivate tobacco, but
- was not committed to "Toby" specifically.
Although he does not appear to complete the riddle by giving an internal explanation for "tobacco", the digression on pipe-weed already being so long that he had to remove it from the main text, I think it is plausible that he planted Old Toby as part of his thinking through the language problems. The name of the character is a backformation from wanting to use Old Toby as the name of the product, which itself is more readily explainable in terms of Tolkien's interest in language than his interest in pottery.
He may have been additionally inspired by the Toby jug but as far as I know there is no trace of this in his writing, which suggests that even if this was something that triggered the idea, it wasn't something he considered to be important in the end. If he cared about the connection then he could have made it more explicit in all sorts of ways in-text. For example, we see various barrels, pouches and jars to hold tobacco, but none of them are associated with the word Toby or are head-shaped. When Merry says (in the chapter Flotsam and Jetsam), "It is Longbottom Leaf! There were the Hornblower brandmarks on the barrels, as plain as plain", Tolkien might have had him say that Old Toby's face was the brandmark - but he didn't. He could also have asked translators to preserve "Old Toby" as an exact phrase, but he didn't do that.
There is no harm in the thought that Tolkien might have been reaching for a spare pipe out of his Toby jug at a key moment of writing through all this, but in my view that doesn't really explain anything about the fictional character or his place in the legendarium.
The Problem of Tob-
In the published The Lord of the Rings, "Old Toby" is mentioned in the prologue as both a brand of tobacco, and the nickname of its hobbit cultivator, Tobold Hornblower. This passage, originally part of Merry's narration at the destruction of Isengard, was transposed to the Prologue in 1948 (estimated by Christopher Tolkien in The Peoples of Middle-Earth, 1996). The character's name appeared in various drafts as Elias Tobiasson, Tobias Smygrave, and Tobias Hornblower (ibid.), as Tolkien experimented with different names that might be abbreviated to Toby. He moved to Tobold because of the distracting collision with the Hebrew name Tobias, as with using Samwise rather than Samuel.
The tob- sound was therefore something which Tolkien considered essential, even as everything else about the name changed. Internally, Tolkien associated it with a "real Hobbit-name", Tobi or Zara-tobi. His note to translators instructs them to replace the brand name Old Toby with "whatever equivalent of Toby is used for the personal name", which I think means that he is not requiring them to use precisely "Toby". But given that the English name comes from Hebrew, also received into other European languages, Tolkien is probably expecting that "whatever equivalent" will be the local version of Tobias, and something starting with those same syllables. From all of this, I conclude that Tolkien was keen on the sound, but not fixed on the exact word. He could have insisted upon it, but didn't.
Tobacco as such is only named once in the book - it's called "pipe-weed" and has various names in the invented languages of the legendarium - owing to Tolkien's discomfort with importing a word of non-European etymology. The precise etymology is contested but he would have regarded it as coming from the New World, in an era that's more modern than what he wanted to evoke (see its entry in the first edition of the Oxford English Dictionary, where it's seen as "quite naturalized" into modern English, after appearing in the 17th century). Similar doubts afflicted the appearance of potatoes, usually called "taters" in-text, as well as tomatoes, tea, and coffee.
Unfortunately for Tolkien, he had accidentally committed to tobacco in Middle-Earth by naming it in The Hobbit. He considered revising this in later editions but did not ultimately feel able (see The History of the Hobbit, John Rateliff, 2007).
"Good morning!" said Bilbo, and he meant it. The sun was shining, and the grass was very green. But Gandalf looked at him from under long bushy eyebrows that stuck out further than the brim of his shady hat. "What do you mean?" be said. "Do you wish me a good morning, or mean that it is a good morning whether I want not; or that you feel
good this morning; or that it is a morning to be good on?"
"All of them at once," said Bilbo. "And a very fine morning for a pipe of tobacco out of doors, into the bargain. If you have a pipe about you, sit down and have a fill of mine! There's no hurry, we have all the day before us!"
There are a few other spots where it occurs in the narrator's voice as opposed to character dialogue, which are less problematic. The one instance in the main text of The Lord of the Rings is for Merry, again at the ruin of Isengard, with "He produced a small leather bag full of tobacco."
Tolkien certainly gave thought to why the plant would be found in his version of the Old World, being conscious that it was an anomaly. Merry's presented theory is that galenas came from Númenor, and was brought northwards from Gondor after the time of Elendil. Notably, we hear the name westmansweed for it, and of course a "Man of the West" is a Dúnadan or Númenórean. So there is a vivid internal history for the plant and its names, albeit stopping short of hanging a lampshade on the word "tobacco".
Riddles in the Dark
If we wanted to follow this etymological puzzle, we could say that the internal etymology of tobacco is from the name Tobi, plus the adjectival suffix -cund. In Old English, which is identified with the language of Rohan and akin to the hobbits' language, -cund denotes origin, derivation or likeness to the root-word. For example, godcund means god-like or sacred, and feorrancund means of foreign-kind. Then a plant that is tobicund means "coming from Tobi", "of Tobi-kind", and tobacco might be its evolved nominal form. (This actually doesn't quite work in OE, but there are plenty of compounds of this type in Germanic languages generally.)
Alternatively, Tolkien also presented "Tomacca" as the original name of Tom Cotton (Sam's father-in-law) in draft - it was "Tolma" in the published Appendix F - indicating some toying with an "-acca" element, which would give another avenue for that sound-combination to appear in Hobbit nomenclature.
This is all speculative but I think it is in the spirit of Tolkien's language-thinking, and roughly consistent with the traces we do have of him working through the language issues. His invention and revision of hobbit names is indicative of trying to reconcile these multiple factors, and represented a major focus of his effort - just look at those hobbit genealogies! Ultimately, I think that any analysis of why "Toby" arises in the text has to be grounded primarily in linguistic considerations of this kind: this is what he was mainly playing around with.