Cleaning and Shining an Old Kitchen Floor?

I have question about rubberized based with thin wood looking top kitchen flooring. I don't know if it's actual rubber or just what type of material? This floor has been in a kitchen for close to 35 years or so and has had approximately the same amount of years of Mop & Glo applied to it from the time it was put in. As far as the owners know has never been stripped.

I am looking for a way of keeping the floor clean, and give it a shine "naturally" without leaving any type of layer or residue behind that could be scratched, marred or scraped by seats or chairs sliding across it or even a large dog romping and running across it and skidding leaving its nail marks on it. Is there any such way cleaning and shining a wood looking for that wouldn't leave a layer behind that a dogs nails would leave scrapes and scratches in it?

By Clayton