Yes, almost certainly Virginia Creeper, Parthenocissus quinquefolia. A grapevine relative, although it doesn't have edible fruit. The birds like it, though. And certainly not poisonous to the touch. WW Here is a link that might be useful: Virginia Creeper

What people do is cut the vine at the base and leave a couple leaves. Then they paint the leaves and the cut stem of the vine with roundup. Roundup is then transported by the vine to its roots where it kills the roots of the vine. It does not kill neighboring roots, so it is safe to use this way. The roundup must not touch any green parts of the azalea or other desirable plants, since it enters the green tissue and will kill that plant also. With vines, it is just about impossible to spray without hitting neighboring plants. Another alternative to brushing is to put on a cotton glove over a rubber glove and moistening the cotton glove with roundup and wiping the roundup onto the stub of vine under the desirable plants, avoiding any contact or dripping on the desirable plants. With nasty vines, they use strong concentrations of roundup or even the straight concentrate. That is quite risky since it will kill anything green it touches. If you haven't used roundup before, it takes about a week to 10 days to kill a plant. However if you get some on a desirable plant, you only have a couple minutes to wash it off. Roundup is an interesting material. I have some large trees next to my driveway. The lower branches obscure my view of coming cars. To defoliate these branches, I spray these branches with roundup. It doesn't kill the tree. Since these branches are such a small part of the tree, it just kills the portion of the roots that are feeding these branches and these branches die and the tree remains healthy. Many ivies and poison ivy have waxy leaves are are harder to kill with roundup. It takes a stronger concentration.

It is possible, and you are doing the best you can. It is possible that when all the stuff was ground up for mulch you got seeds in there and started it all up again. You've also got some really, really stubborn weeds; I once forecast that Virginia Creeper would bring about the end of civilization. You just have to keep after them, as you have been doing. Keep them cut down as much as you can, so they don't go to seed. As you have already figured out, premixed RoundUp doesn't do squat. It sounds like you've got mostly woody weeds, so either use RoundUp at about a 2-to-1 dilution, or get some brush killer, which is, I believe, stronger. Once the mulch starts decomposing, it will be easier to dig up some of the bushes. You didn't say where you live, so I hope that you are in a state where a good cold winter will give you a bit of a break.

Ken... lol... I don't even mow my yard. Generally speaking, I care not a wit about what grows where unless, like this vine, it can take over. Therefore, as long as it is in the yard, and has little practical purpose for being there, instead of giving it space to do its thing, I'll cut it back with the swing blade. Then I'll add it to the garden beds or the compost pile...just as I do with all the grass and weeds already in the yard. Since it does have the practical purpose of being an interesting type of beauty and, as my research has shown, attracts butterflies and hummingbirds, it will continue to be allowed free reign in the treeline. My conception of beauty and desirability is much different than many... especially lawn-obsessed Americans. :-) For example... last time I cut the yard back, I found a few wild blackberries and a few sassafras trees encroaching from the treeline. I let both of them stay right in the front yard. I make my own wine, beer, and soda. Sarsaparilla is now on the drawing board! Woohoo!