Can you stain or paint an old linoleum light colored floor?

Yes, Rustoleum makes a floor and porch paint. make sure your floor is very clean of debris and dirt. Any little divots or cuts in the floor you should use some caulk and smooth them out. Put a coat of primer on there first, let dry, then paint with your favorite color. After it dries well, top with a couple of coats of polyacrylic.

https://www.viewalongtheway.com/2013/01/how-to-paint-vinyl-or-laminate-flooring/

Note: The floor can be slippery if you poly over it. It's even slippery just using Porch and Floor paint.

You can prime and paint your floor. Make sure that it’s very clean before painting. If you use semi gloss and let it cure before walking or moving any heavy furniture. Curing may take at least a week. Check the paint for cure time.