Is the Book of Life the same as the Lamb’s book of life and how could we as saints of the age of grace be written in it since this age was a mystery?

I will have to give some thought to your question. My first thoughts however is that since both are called “The Book of Life” that they would be the same. I have always thought that when Israel was in covenant relationship with God they were born into this covenant (especially if circumcised at 8 days). Therefore their names were automatically entered into the book of life. The indications from Moses when they worshipped the golden calf is that their names would then be taken out of the book. Since we are a new creation, our names do not go into the book until we believe the gospel where they can never be taken out (Philippians 4:3). Then there are those in Revelation 17:8 whose names never were in this book – most likely Gentiles whose names never got in.