Yes. You'll see them every where in Western NY State, where I live, along with swallow wort, garlic mustard, common buckthorn, and other invasive alien species. Search and destroy is the best approach to these. FataMorgana

Perhaps Rhus aromatica Here is a link that might be useful: Rhus aromatica

Looks like Myrsiphyllum asparagoides, Florist's Smilax.

Looks like one of the dastardly shrub honeysuckles (Lonicera spp.) to me, probably winter or fragrant honeysuckle. That one is known as Lonicera fragrantissima. It will have had small creamy white flowers earlier in your season, from which the red fruit are now forming. The new stem growth extensions with purplish color is the give-away ID to me. All the honeysuckles are oppositely arranged in leaf and branching, making them easy to separate from many other native species (except viburnums, buckeyes, and dogwoods, of course). The behavior of "...the shrubby undergrowth is almost entirely this shrub..." fits with how many shrub honeysuckles dominate the understory of woodlands when they get the chance. The term that fits is invasive. If you are interested in reducing or eliminating this effect, then you have to reduce/eliminate the flowering so that you don't have massive new fruit production every year. That means repetitive cutting down of the plants, especially right before or during flowering, or resorting to total removal or pesticide application.