Legally, the book is useless for the bard. How does this manifest itself at DM?
A shadow book contains only traps:
Your boss gives you a book called the Shadow Book. When you get this feature, choose three rollovers from the list of spells of any class.
If this warlock also invokes the ancient secrets book, he can also add rituals to the book:
You can now include magic rituals in your Book of Shadows. Choose two first-level spells that carry the ritual tag in the spell list of any class (both do not have to belong to the same list).
Since a class can only read spells from scrolls and so on. in his class list, this would mean that a Bard, at best, could only read rituals and misinterpretations that are already in Bard's list of spells.
However, the bard could not use any of Bard's spells in the book as a Bard is limited in the number of word traps and spells he knows (as in all spell casting classes).
On the other hand, a DM could treat the spellbook as a scroll of spells and allow the Bard to launch the contents of the book while he was on it. But that would mean that the Warlock could not then cast those spells, because the pact also says:
As long as the book is on you, you can launch these cantrips at will.
And Summoning: With your book of shadows in hand, you can cast the chosen spells in the form of rituals. You can only cast spells as rituals unless you have learned them otherwise.
This means that without the book, these spells can not be cast. Once the bard made the spell to the warlock, he would lose access to those spells. The net number of known cantrips and rituals in the group would remain constant in both cases (unless the Warlock naturally knows one of the ritual spells in the book).
How the DM decides to handle the mechanisms from a traditional point of view is his choice. The bard can identify the type of magic for each spell (this is in accordance with the rules of the Detect Magic spell), but this has no real value. Maybe he / she is unable to read the book writing at all because the text appears as swirling and shapeless shadows, etc., so the nature of the magic is all that that she can glean.