The #1 culprit of damage to marble floors is dirt, (grit, sand & dust). The best treatment for your floor is a dry, untreated microfiber dust mop or vacuum, used every day when possible. Damp mopping is recommended on a weekly basis and whenever there are visible spills.
When acid (vinegar) comes into contact with marble (calcium carbonate), it causes a chemical reaction. The result is called etching. Natural stone sealers penetrate the stone and do not prevent this natural reaction. If you do in fact have a polyurethane coat on top of your stone, then you have a different problem altogether. If the stone is etched, we can refinish it.
If your floor is not deeply scratched, surface polishing may bring back an acceptable shine. However, existing scratches may remain. If your floor has worn badly, a complete restoration is warranted. This decision should be made between you and us.
Rugs are a good idea to catch dirt and grit. However, there are a few things to keep in mind. It is important to use rug holders designed to go under rugs to keep them in place. Sliding rugs are not only dangerous, they are a constant source of new scratches. Do not use rugs that are backed with rubber or latex.