Mulched privet lastfall…
Yes, it looks like you have Common Buckthorn which also called European Buckthorn, Ramnus cathartica. It is an invasive species and should be removed. According to the City of Toronto Urban Forest Management website, you can dig up or hand pull smaller stems. Large vines need to be cut at the base repeatedly. Cutting repeatedly will eventually starve the roots of nutrients (no leaves for photosynthesis).
Here is a more detailed explanation for the Ontario Invasive Plant Council: When the soil is moist, small plants up to 1 m (3ft)
in height can be pulled. As Common Buckthorn grows, it puts down a deep root system very quickly (dependent on soil conditions) and sends out lateral roots that can be as long as 2 – 3 m, which will also need to be removed. Larger plants can be dug out, or pulled out using a weed wrench tool. Re-sprouting can occur unless all the roots are removed, so you need to keep an eye on the area and pull up any new stems you see. If pulling takes place in the fall, care should be taken to remove and contain branches with berries prior to pulling. This plant also has thorns, so wear gloves. Because of the thorns, you should wear gloves. Pulling is extremely difficult in clay soils. It works best in fresh-moist loamy sites or when the soil is moist or wet. Don’t put common buckthorn in home compost or out for city yard waste collection, but in your garbage bin or bags for city collection.
Following the link below will lead you to a 35 page document on buckthorn removal, if you wish more information.