I remember there was a separate book for D&D 3.5 describing all about Forgotten Realms. My question is - is there a book which describes it for 5e?
The only official book I can think of that describes Forgotten Realms lore in detail for 5e is the Sword Coast Adventurer's Guide.
The first couple of chapters are all about the Realms, it's history and geography, etc, before following up with how the classes and races from the PHB fit into the Realms specifically.
It also includes a few Realms-specific class archetypes such as Bladesingers and Long Death monks, as well as being the only place (that I'm aware of) where the stats for Duergar as a playable race exist.