I have a mystery bush that came up about two years ago next to my Lincoln Rose. Thought it might be a new rose bush that just came off that one, but it started flowering white small flowers this year. It has been growing for three seasons now. I am in Southern Indiana if that helps.
Looks like Rosa multiflora, which is invasive in your area. You might want to think again about keeping it!
Is this possibly just the root stock sending up shoots? If so, the growth could take over and weaken the grafted plant.