In the descriptions, they each say things like "You can use this book as a spellbook" and "It functions as a spellbook for you", so I expect that spells can be added to them. Am I correct?
This is correct.
The Wizard’s spellbook feature says:
Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it.
When the Tasha’s spellbooks say “It functions as a spellbook for you”, they are referring to the Wizard’s spellbook feature quoted above.