In the last shot looks like Japanese honeysuckle - as I suspected it might be - which is a noxious weed in eastern North America anyway. So the thing to do is to keep working on it, get it off your site entirely.

And remove any future appearances of replacement seedlings when the plants are still small.

correct.. first ID the H ... pix before you remove it next time.. lol ..

then eradicate it.. if its the bad one ...embo seems to ask for a better pic to confirm ...

you can cut it to the ground and treat the cuts with undiluted round up or stump killer.. so as to not have to also dig up the lilac at the same..

also.. google rejuvenation pruning of flower shrubs.. and get the L back in better shape ...

It's definitely the "bad one" but for the moment it serves a purpose since it provides much needed privacy. I won't get rid of it entirely but I wish there were a way to contain it - it's doubled in size in a year and that lilac is definitely suffering. If only it were alone on its side of the garden... It sounds like it'll be an ongoing battle with the honeysuckle, as I'm learning more about the garden. I have another ongoing battle with the Virginia creeper . . . Thank you both for your responses!

Since it is vigorous and flowers on the new growth of the same season Japanese honeysuckle can be cut back every year in early spring and still make a full presentation later the same year. If you want a vertical wall of screening growth cut the side branches back each year but leave a framework of permanent main stems to continue the screening effect.

Otherwise where it has become a mess the entire top can be cut down to about 2 ft. tall in early spring and allowed to start over. Again it is likely to bounce back so fast flowers will be seen the first summer after this is done.