Try the name that plant forum which is more active and has some real whizzes at plant ID.

It might help if you add a photo of the lilac and vine growing together.

Looks like Euonymus fortunei, an invasive species in much of the US. Find where it comes out of the ground and cut it through. Either keep cutting when it resprouts or dab some undiluted weedkiller on the cut surface. The vine on the bush will die and can be carefully removed.

Euonymus fortunei 'Coloratus' - Plant Finder '' Wintercreeper (Euonymus fortunei) has been identified by a task force of the Missouri Botanical Garden as one of the top twenty plants known to be spreading into native plant areas and crowding out native species in our region. Wintercreeper spreads both vegetatively and by animal and water dispersed seeds. Because of its known invasive tendencies and difficult to control dispersal mechanisms naturalists recommend against planting this plant.''

'''Garden Uses

Where not deemed invasive it can be a versatile
ground cover for sunny or shady areas in the landscape. Also effective
as an edger along paths or sidewalks, as a slope cover where it can also
provide erosion control, as a foundation planting. If used as an
ivy-like climbing vine for covering walls, chimneys or fences, it more
easily flowers and sets seeds that can increase its invasive spread.''