I believe the vine is bindweed and the other plant is a mulberry.
hortster

are they easy to remove? Mine came back after 3 years.

Both plants have root systems from hades. Constant removal each time they rear their ugly heads is the best answer. Get all that you can each time. Tenacity!
hortster

Bindweed can have an extensive root system that will allow regrowth if not completely removed but birds eat the seeds and drop them and these seeds germinate and grow and that is more likely what you had after 3 years, reseeding.
Mulberries grow from seeds so if someone is seeing Mulberry trees growing in their garden there is one around, somewhere that the birds eat the fruit of and drop the seeds that germinate and grow. If new growth from those roots is not cut off fairly quickly the roots can grow enough to allow new growth from those roots, but diligence in keeping any new growth cut off which then deprives the roots of needed nutrients so the roots die is necessary.