Do the books ever specify whether or not another ship of this make exists? Or is it truly unique? Maybe a long time ago in a galaxy far far away manufacturers made single one-off units, but here that's kind of rare for something you'd assume to be pretty costly to build.
The Millenium Falcon is not unique; it was a model YT-1300f Corellian light freighter. This implies that there are other similar ships, including other variations on the YT-1300 model.
You can find a lot of information about the Millennium Falcon in a companion book, Star Wars: The Force Awakens: Incredible Cross-Sections. The model number also appears on-screen at least once: in Star Wars: Rebels episode "Fighter Flight" on a poster.
Note, however, that the Millenium Falcon has been highly customized; some of the custom work is mentioned in The Force Awakens. It's not clear just how much of the ship is stock and how much is custom.
If you're willing to go outside Canon into Legends, the model number of the ship, and other similar Corellian light freighters, can be seen in some of the video games, and there is an entire novel called Millennium Falcon that details its history. The ship was serial #YT-1300 492727ZED, one of many such vessels, but from the beginning the ship was shown to have some "quirks" to distinguish it from its counterparts.
In Legends, the ship was also destroyed and rebuilt from scratch at least once, so it's arguable as to whether it's still the same ship or not.
As has been mentioned, the Millenium Falcon is a highly modified YT-1300f freighter.
If you look carefully in Attack of the Clones and Revenge of the Sith, you can see other ships of the same class:
Image source: 10 Hidden Details in Star Wars Movies, Screen Rant
There is an official Millennium Falcon Haynes Manual. The first part of the book discusses all the models of the YT-1300 Corellian Freighter which are available for purchase and later on it goes into specifics about the actual Millennium Falcon and its modifications.
Think of the Millennium Falcon like this:
There are lots of Nissan Skylines, but this is a highly modified one (within the "universe" of the Fast and the Furious films, as well as in the real world) and is therefore unique. There is only one Millennium Falcon. Only one. But it is a modified version of a pretty common freighter. Indeed, in the old Legends game Star Wars: Rebel Assault II: The Hidden Empire, you have "The Corellia Star":
To give a fully movie-canon answer, in The Force Awakens:
After Han informs Rey that they are on the Millennium Falcon we have the following line 1
REY: (surprised) This is the Millenium Falcon! This is the ship that did the Kessel Run in 14 parsecs!
Since Rey is an expert on salvage and ship models, and since she knows about the Kessel Run, we can reasonably infer that:
- She knows what a YT-1300 Correlian freighter looks like.
- She knows the Millenium Falcon was a YT-1300.
Yet she didn't suspect that the ship in the junkyard was the Millenium Falcon. Therefore, it must be that the Millennium Falcon is not unique.
1 I'm going from memory, if someone has the exact quote from the novelization, please update
In Star Wars, Han says 3 things that hint at its modifications:
It has better-than-expected sublight and lightspeed engines than one would expect out of that kind of freighter. ("She's got it where it counts, kid.")
It has a lightspeed engine that can shorten distances next to gravity wells ("It made the Kessel run in under 12 parsecs"). In the novels, this is taken to mean that a better lightspeed engine can go closer to stars and take shorter paths than would be possible with a lesser engine. Very useful for getting away from Imperial Star Destroyers.
It has hidden cargo bays that apparently are VERY good at fooling scanners. ("I use them for smuggling. I never thought I'd be smuggling myself in 'em.")
It's not clear if he won it that way or modified it himself. But all this implies that it's a standard cargo ship that has been specially modified for smuggling.
I hope I am doing this correctly, to point to an existing question-and-answer. Not literally a duplicate, but in effect the substance has been covered.
The sense of this question was addressed in the discussion of another that was answered at 'What is the Millennium Falcon’s hull and structure made of?'. It involves the Haynes Manual, as has been mentioned here.
The essence is that the Falcon is no more a one-off in production terms than any other YT-1300, in the sense that multiple design options were so flexible to start with that the term doesn't really apply. All of the YT-1300s were sort-of one-offs, therefore, but this particular one seems to have become even-more-unique (tm) over time since manufacture.
In Heir to the Empire (part of the Empire trilogy by Timothy Zahn) the Nogrhi commandos use a YT-1300 to try to trick Han and Leia so they can kidnap them. Han expresses genuine surprise that they've been able to find the same make and model.
“Good question,” Han said grimly. He leaned out again, taking a hard look this time … and when he ducked back under cover there was a sardonic half-grin on his face. “Simple answer: that’s not the Falcon.”
“What?” Wedge asked, his jaw dropping a couple of centimeters.
“It’s a fake,” Han told him. “I can’t believe it-these guys actually dug up another working YT-1300 freighter somewhere.”
Wedge whistled softly. “Boy, they must really want you bad.”
In the Star Wars: The Essential Guide to Vehicles and Vessels factbook (published in 1996) the section on the Millennium Falcon says that
the Falcon appeared to be no different from the thousands of other Corellian Engineering YT-1300 light freighters plying the star routes of the Empire.
So from that it would appear that it's a mass-produced vehicle.
Also, the Outrider piloted by Dash Rendar (in various EU Novels and games) appears to be a later model in the YT series of ships as this is listed as a YT-2400 light freighter.