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

I noticed a long time ago that the Flag of Unity doesn't give +400 mana at all. I checked this in a tournament with Mar. The other bonuses seem to be working fine. Is this a bug? Or am I imagining things?

The Flag only gives a bonus when a hero visits it for the first time and after the hero has been in battle. The fact that it has been visited is visible in the tooltip. This also applies to other one-time mana bonuses. However, there is also a downside - after each battle, you can visit the same flag and receive mana. This is more of a feature than a bug - an additional reward for winning a battle.