Looks a lot like a vine called Virginia Creeper, used in landscapes. Makes a very nice vine on walls or trellises. The leaves turn a beautiful red in the fall, then goes dormant for the winter. It grows an extensive root system and is very difficult to get rid of once well established. However, if you use some full strength glyphosate, not the ready-to-use mix but the stuff that is 40% or more active ingredient, you can use a cotton glove or paintbrush or sponge and coat new sprouts when they come out of the ground. This allows you to miss getting your holly by accident, and it will take the vine out but may need a couple applications as regrowth appears. The Roundup won't hurt the holly as long as you don't get any on the holly leaves even though their roots are close together. Your other alternative is to continue doing what you are doing now--keep cutting it off everytime some growth appears out of the ground to prevent any photosynthesis and gradually starve the roots out.

^^ The foliage is smooth and has not much fragrance when rubbed. And here is another shot. I guess I'll figure it out for sure whenever it starts blooming?

Source plant probably visible somewhere in vicinity.

Ungnadia speciosa, Mexican Buckeye