Nintendo, Sony or both ? 

Giant investors you have never heard of.

Nintendo. They are the only ones who dominated both home consoles and handhelds. And their 1st party IP's are unmatched.

Oops, I'm sorry, I meant only between Nintendo and Sony haha

Hmm, Nintendo imo but compared to the whole industry Sony and Nintendo do not come close to where the real money is.

No one company dominates the industry. Sony has done the most domination in terms of home consoles sales and attached software.

Nintendo is like the comeback kid that gets up hard and in your grill just when you thought it got tko’d. They’ve been around and have a shady history.

Microsoft just keeps throwing money bags at everything and acquires the games I want personally.

Neither. Domination implies steady influence over the other and neither has been remotely consistent throughout their history.
Mobile gaming producers/investors would be a more suitable answer.

In what category? In hardware sales? Software sales? Revenue? Services?

Tencent

Nintendo and Sony are roughly tied. Tencent and Apple are also around the same level, and Microsoft and Google are not too far behind.