The demon's apprentice

Void

As the fight went on, with Tanjirou fighting the boar headed demon slayer and  the mist hashira, who is still very injured, Kokushibo finally made up his mind.

Yes, he did infact care about Tanjirou.

Even though Tanjirou did look a lot like him, he still wasn't. Tanjirou was not Yoriichi. Yoriichi died long time ago. Tanjirou is a young boy under his care.

Has Kokushibo maybe imagined Tanjirou as Yoriichi to fill in the void in his cold heart?

No. He doesn't think of Tanjirou as a reincarnated version of Yoriichi, and never will. But somehow, Tanjirou still does fill in the void that Yoriichi left behind.

As the mist hashira slashes dangerously close to Tanjirou's neck, it's suprising how he's able to move at all, Kokushibo feels an emotion he hasn't felt in centuries.

Anger.

Kokushibo feels rage build up at the thought of Tanjirou dying, and he realizes how foolish he is. Anger is a human emotion. Tanjirou isn't related to him and just like Kokushibo, he just another demon serving Muzan until they get beheaded. So there's no good reason for him to feel angry at that moment. Same goes for the moment where the mist hashira, named Muichiro, talked about Tanjirou's past, his human life. The thought of Tanjirou regaining his human memories and hating Kokushibo is another thought that makes Kokushibo's blood boil. He's an idiot.

He's getting weak. Anything remotely human in a demon makes them weak. Any type of relationship that has even a little bit of family like bond, weakens everyone. Just like Tanjirou's blood demon art. It makes the victims see a dead relative. It makes them vulnerable, an easy target.

That's not true.

Was that Kokushibo talking to himself or something else buried in his already messed up mind? Who knows.

While human emotions makes demons weak, one human emotion can make them even more powerful. Fondness. The need and want to protect a loved one can make someone even stronger.

Kokushibo has heard all of those cliché stories. Where parents are granted supernatural powers when their child is in danger. In those moments the parent isn't thinking on any logic, just listening to the absolute need to save their kid. Get the kid from under the tree. Push them from the oncomming carts way. Doesn't matter how or what you do just save them! He's heard them all.