While there is scant information, the Book of Life is mentioned throughout the Bible, and it is called by several titles and always in the singular sense:

God calls it "My book" (Ex 32:33).

Jesus calls it "the book of life" (Rev 3:5).

Angels call it "the book of life from the foundation of the world" (Rev 17:8).

Prophets call it "the book of life" (Ps 69:28; Phil 4:3), or simply "the book" (Dan 12:1), or descriptively as "the book of remembrance" (Mal 3:16).

The apostle John, after ingesting the little book from the strong angel and becomes endowed to prophesize to the world (Rev 10:10-11), calls it "the book of life of the Lamb" (Rev 13:8) and "the Lamb's book of life" (Rev 21:27).

When reading some references, there is a suggestion that the names of Believers are added to the Book of Life when they come to faith:

Now at that time Michael, the great prince who stands guard over the sons of your people, will arise. And there will be a time of distress such as never occurred since there was a nation until that time; and at that time your people, everyone who is found written in the book, will be rescued. (Dan 12:1)

Indeed, true companion, I ask you also to help these women who have shared my struggle in the cause of the gospel, together with Clement also and the rest of my fellow workers, whose names are in the book of life. (Phil 4:3)

All who dwell on the earth will worship him, everyone whose name has not been written from the foundation of the world in the book of life of the Lamb who has been slain. (Rev 13:8)

"The beast that you saw was, and is not, and is about to come up out of the abyss and go to destruction. And those who dwell on the earth, whose name has not been written in the book of life from the foundation of the world, will wonder when they see the beast, that he was and is not and will come." (Rev 17:8)

In the daytime (for there will be no night there) its gates will never be closed; and they will bring the glory and the honor of the nations into it; and nothing unclean, and no one who practices abomination and lying, shall ever come into it, but only those whose names are written in the Lamb's book of life. (Rev 21:27)

However, when understanding this issue within the whole context of the Bible, it appears that the names of all of mankind are initially written in the book "from the foundation of the world" (Rev 13:8; Rev 17:8) and removed when they fail to come to faith. This provides a context to Ephesians 1:4 - just as He chose us in Him before the foundation of the world, that we would be holy and blameless before Him.

But now, if You will, forgive their sin—and if not, please blot me out from Your book which You have written!" The Lord said to Moses, "Whoever has sinned against Me, I will blot him out of My book. (Ex 32:32-33)

He who overcomes will thus be clothed in white garments; and I will not erase his name from the book of life, and I will confess his name before My Father and before His angels. (Rev 3:5)

The Book of Life appears to be written in the presence of God and its purpose is to list those who fear the Lord and who esteem His name, distinguish the righteous from the wicked and indicate who will serve God and who will not.

Then those who feared the Lord spoke to one another, and the Lord gave attention and heard it, and a book of remembrance was written before Him for those who fear the Lord and who esteem His name. "They will be Mine," says the Lord of hosts, "on the day that I prepare My own possession, and I will spare them as a man spares his own son who serves him." So you will again distinguish between the righteous and the wicked, between one who serves God and one who does not serve Him. (Mal 3:16-18)