Do you know why these are used as singular forms?
keep an eye on that, and use the other eye to keep doing whatever you are doing, is the idea. If you keep your eyes on that (which can be said as well), then you are supposed to look only at that thing, and nothing else.
No special reason; just idioms.
I'm not positive, but I've always felt that these expressions indicate keeping track of something that's not currently a major concern, or at least not your only concern. For instance suppose that my doctor tells me that my blood pressure is a little high and I need to keep an eye on it. That means I'll check it several times a week. But I also have to work and pay attention to all other aspects of my life. I can't take my blood pressure all day and all night. Or suppose I want a new coat. I'll keep an eye out. This means that I'm not going to the store just for a coat. But if I see one on sale that I like, I'll consider buying it. On the other hand, when you are learning to drive, the instructor doesn't tell you to keep an eye on the road. It's "eyes on the road", meaning both eyes and full attention. No eye for your cell phone.