I couldn't find a good review of this book, and I believe I understand why ... Book 1 is 670 pages, Book 2 is 790.
In the intro there is a reference to a fairly well known tale of a woman who read Hamlet and remarked "I don't see why people admire the play so, it is nothing but a bunch of quotations strung together". That is much less true of this book, however the lack iof general public comprehension of what they are reading is probably similar.
This book is much more history than "Shakespeare", although it does have a significant number of the more famous lines in it. It focuses on the context of the period, what real or literary/legendary person the lines are likely referring to, and why.
Shakespeare was writing for both the common man as well as the wealthy aristocracy and royalty. What a "commoner" needed to know about Greek, Roman, Italian, the Bible, English history, etc gives a little insight as to why even the "educated" our day, being "experts" of only their iPhones and the latest Netfix binge watch, have a hard time understanding why throwing a trillllion dollars into an "Inflation Rediuction Act" might cause some brows to raise. "Common Sense" is far from common today.
Also, if Shakespere had put in any obvious snark like I just did, he would likely be "cancelled" by literally losing his head. He was marvelously subtle with his little jabs.
On Page 9, a helpful map of the Roman and Greek gods, with their role is presented. To cover a few of the more popular ones, in Greek we have Zeus, chief of the gods, Athena, goddess of wisdom, Ares, god of war.
In the Roman version we have the corresponding Jupiter, Minerva, Mars ... you need to understand these references to keep up between the Roman and Greek plays.
A very helpful page for those of us that don't have the memory of our youth, and received at best a very cursory understanding of the ancient world. Volume 1 covers the Greek, Roman, and Italian plays. The Italian plays are probably the most familar ... Love's Labor Lost, The Taming of the Shrew, The Two Gentlermen of Verona, Romeo and Juliet, The Merchant of Venice, Much Ado About Nothing, As You Like It, Twelfth Night, All's Well That Ends Well, Othello, Measure For Measure, The Tempest.
The book is loaded with inline cross references to where subjects are covered in the other plays, in order to better understand what is being covered ... the audience at the time of writing had a common understanding of the world they lived in, including the history, and had good memories ... uncluttered by shallow media entertainment.
Asimov made a number of desisions as to how to present this vast environment to the modern reader, largely unaware of the world of Shakesphere. I think of it a bit as a kalidiscope of "worlds", with sometimes definite and sometimes completely fictional references to real, mythological, current, recent historical, fictional charachters invented for the story, etc.
Asimov is in a way trying to put us into the Shakespere world ... a BIT like todays "Marvel Universe", "Star Trek Universe", "Star Wars Universe", etc Think of a reference to "Captain Kirk" 500 years in the future. Yes, I know, that is shallow ... maybe "Winston Churchill", or maybe "Dostoevsky" would be a better example.
The big differece is that while Shakespere is "fictional entertainment" it has much more connection to thew reality of the time. Maybe something like "The Crown" today.
Do I recommend the book? To the common reader of today, I really can't, because they are likely to just be frustrated and lost. Certainly there are a decent number of people FAR more qualified than I to read and enjoy the work. Perhaps I'm an arrogant pessimist, I just don't think the audience to actually read it is very wide ... it does however look good on a shelf, unless it is full of tabs like mine is.