Cleaners Turn Floor Tile Bluish Color?

I'm pretty sure I have linoleum tiles in my bathroom (though they may be vinyl, not sure how to tell the difference). They're an off white color and probably from about the 70s or 80s. Every time I try to clean or scrub them, they start to turn a blue grey color and no matter what I use (I've tried dish soap and water, vinegar, baking soda and water, Magic Erasers, bleach and water, and a general all purpose cleaner) or how much I scrub it seems to just spread this weird blue grey color around (sometimes it even seems like scrubbing makes it worse/darker).

Does anyone know why this is happening or what to do to clean it up? It's very annoying, it makes the floor look grungy and dirty even though the floor is clean.

By Chris