Technically, the flange is the whole funnel shaped fitting. The metal ring is part of it used to hold it in place onto the subfloor. If you do cut it off, they do make repair rings designed to fit into the groove that's holding it now. THey're split so you can easily close it into that slot.
Depending on your access, it might be better to cut out the whole flange and then install a new one once your new floor is ready for it. Those flanges are designed to be install on TOP of the FINISHED floor. Numerous ones are not installed that way, but that means you need more care to get a good seal and keep it intact. Plumbers prefer not to use those with a painted steel ring or one made entirely out of plastic. If you cut off the existing ring, select a SS replacement. OFten, the painted steel ones end up rusting, which isn't a great situation.