The classifications do generally match up. Makes and models come and go, but typically you would find that a model that is classified as a fullsize for example at Hertz would be the same at Dollar/Thrifty.

As far as sharing fleets/pooling vehicles, they are typically separate. There tends to be a lot of variability in the fleets and I don't believe there is generally a concerted effort to push the older/lower trim vehicles to Dollar/Thrifty, but YMMV depending on renting location and just what's available at the time you arrive. You certainly could get something brand new or almost brand new at Dollar/Thrifty just as you might at Hertz.