This is actually one of my favourite weird trends in the history of weaponry! I can't really comment on the likelihood that Morozini ever had to use his gun bible - Venetian politics is not my area - but I can talk a bit about how that gun got there! First things first, though, for those who may not be familiar, this is the weapon in question.
Morozini's gun Bible used a flintlock trigger system - probably one of the most influential methods of firing a gun ever invented. However, this type of device dates back a little further to the origins of the wheellock gun mechanism in the early sixteenth century. Wheellocks forever changed the way guns were used and designed for one very simple reason: wheellocks removed the need to use an open flame to fire a gun. Back in the Middle Ages the way you made your gun shoot was by literally lighting the gunpowder on fire, ideally this was done with a slow burning match (more like a rope than a modern match), by attaching these matches to a trigger system the matchlock gun was invented. The wheellock instead used a rotating piece of steel wound in place via a spring to scrape against a piece of pyrite (this is important, don't use flint in a wheellock pistol!) to create sparks which would light the powder rather than a direct flame.
This meant that gunners no longer needed to carry around open flames to shoot their guns. By far the most significant development to follow this innovation was the invention of the pistol - small guns you could put in your pocket are a lot more viable when you don't need that gun to also be a little bit on fire to work. However, more to our interest there followed from this a fad where people basically started putting wheellock guns on basically everything they could. See the following for an abbreviated sample list that will hopefully make my point:
Now, none of the above are strictly on the same level as the gun Bible because they are at least actually weapons, just ones that guns have been added too (probably unnecessarily). What they do show is how good at miniaturizing the wheellock (and later flintlock) trigger systems early modern engineers became. Once you can start fitting guns into the hilts of swords, you can basically put them anywhere, and there was clearly a demand for it. In addition to the weapons above, there are examples of putting wheellock guns into shields, but tragically I couldn't find an example in any available online collections.
Here's the thing, though, none of these weapons were particularly good. Wheellocks weren't unreliable, but they did misfire and they were pretty sophisticated pieces of engineering. They certainly didn't benefit from being smashed into things, which is what would happen with these weapons if they were used in a major battle. It has generally been assumed that these weapons were novelties, a way of showing off ones status, and the gun was only for self defense in a secondary or tertiary capacity. If I'm honest, we're really not sure what's going on with the gun crossbows at all, but they are rad and I'm inclined to think people 400 years ago also thought so.
In terms of how common things like Morozini's Bible were, his book was not unique (although again I'm struggling to find a good image of another example), but they certainly were not common. Things like swords with guns on them are far more common than more mundane items with guns hidden in them. I have seen some other examples, though. In the collection at Grandson Castle in Switzerland they have a pair of eating utensils (a fork and a knife) that have guns in the handles, which the curator Ian Ashdown assured me were more common than you'd expect (but then I didn't expect them to exist at all, so I don't know how common that makes them really). Ian Ashdown also mentioned the existence of gun books, but tragically there were none in Grandson for me to see in person. I also forgot to take a photo of the gun utensils (I was technically there researching crossbows, not weird guns), something I have regretted ever since because none of that collection seems to have been digitized.
I still can't comment on the likelihood that Morozini ever needed to use his Bible (my gut says he probably didn't, but my gut is terrible at providing proper references so I wouldn't take his word for it), but what I can say is that while not unique, items like Morozini's bible or the gun cutlery represented a niche section of a larger fad of putting guns on stuff that was huge from around the mid-sixteenth century until at least the end of the seventeenth century.
For a similar technological fad from the same time period as all this was happening, check out the ballestrini, sometimes called "assassin's crossbows". These were small (hand sized) all metal crossbows that were spanned with a clever cranking system stored in the crossbow's stock. They could be surprisingly powerful little weapons. Despite their name, though, there is no evidence they were ever used for assassinations and in fact seem to have primarily been a toy that rich people used for fun. Kind of like a 1600s BB gun. The Met has a good example of a slightly later ballestrino.
I don't think anyone's written a dedicated book on this, but for general early gunpowder history the work of Bert Hall is just stellar, his Weapons and Warfare in Renaissance Europe: Gunpowder, Technology, and Tactics is a great introduction to the subject.
Dirk Breiding talks a bit about the Met's gun crossbow in A Deadly Art, which would include information applicable to some of the other items discussed in this post.