HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

I have SoD, the latest mod, and in three games in a row, heroes are given books without having a book skill.

I can partially confirm this.

If the first battle after launching the game is played with quick battles disabled, then the hero receives a book after that battle; I'm attaching the save files.

Until the author fixes it, I think it will be enough to create a single-player game and play a battle before connecting with an opponent, then connect through the menu without restarting the game.
I hope it works; otherwise, I'll have to withdraw from the SoD tournaments.