This looks like a privet of some sort, Ligustrum species. If you can clip some flowers, lay them on a plain smooth sheet of paper(preferably in a contrasting color) and take a photo so we can see the stamens, the number of petals, etc. that would be helpful.
In any case, to remove a shrub you cut it at the base and immediately(within 5 minutes or less) paint the cut stems with a brush-type herbicide used for poison ivy, brambles, etc. Active ingredient is either triclopyr or glyphosate.  Ortho, Roundup and BioAdvanced are three common brands, for example. There are others.

Here is a bulletin with other removal options.
https://extension.illinois.edu/invasives/invasive-privet
MSU Extension doesn’t endorse brands or retailers, they are mentioned as examples only.
Please read and follow all precautions and directions on any chemicals used.
And, do not get the chemical on any trees, bark or other desirable plants(shield or cover them until applied chemical is dry, then uncover or remove the shields.

Thank you for the additional picture. This European privet, Ligustrum vulgare, or some hybrid. It is not native. Here are more bulletins about managing it.
https://ohioline.osu.edu/factsheet/F-103

https://mnfi.anr.msu.edu/invasive-species/InvasivePlantsFieldGuide.pdf

Your are very welcome, good luck with your “project”