Simply put, that's because both France and Belgium had a colony called Congo:

• the French Congo, part of French Equatorial Africa, became the Republic of Congo, (Brazzaville-Congo let'say),
• the Belgian Congo (formerly Congo Free State) became the Democratic Republic of Congo (former Zaire, Kinshasa-Congo if you wish)

Both colonies took their name from the Congo river, which ultimately was named after the Kingdom of Kongo.

It's a bit like Niger and Nigeria, both named after the Niger river.

The Belgian and French agents, notably Stanley and De Brazza respectively, had actually been collecting "treaties" and concessions along both sides of the Congo; the Kongokonferenz of 1884-85 (which we know as "The Berlin Conference") was actually prompted by disagreement over the Congo basin, first among all the other emergent disagreements over holdings in Africa. The only actual boundary determined by that conference in fact was between the Congo Free State and French Equatorial Africa (Congo), that single boundary. As /u/dyskolos points out the actual name is taken from the river, not the historical Kingdom of Kongo, which was nearer the coast and partially in Angola. Chapter II of Wesseling's Divide and Rule is still a good read on the geopolitics of it, I think; I don't know if Van Reybrouck (2014) talks about it much.