wait till it leafs out.. and perhaps flowers to confirm such ... if it is a lilac .. its 3 to 6 feet .. TOO CLOSE to the house ... you have to move it ... ken

Ligustrum = Privet, which is what it looks like. Don't know about Texas, but around here they are rather weedy and have naturalized. I imagine digging up a sucker or two would work. They're hard plants to kill. WW

Twigs and buds look like those of a Spiraea.

Ideally I'd think you want a variety of shrubs, evergreen and deciduous, rather than a row of one thing. You can plant perennials in between to give you a bit more privacy during the summer. A rambling rose would be nice. Red-twig dogwood (I like the one with the variegated leaf), yellow-twig dogwood, shrub roses, all kinds of vibernum, hydrangeas, azaleas (The species azaleas have a particularly nice open form), spireas, choysias. . . Underplant with hostas and interesting ground covers. With a little planning you can have something in bloom or displaying berries or colored foliage pretty much all year 'round. If you want to go with native plants consider Oregon grape (the berries are edible), flowering currant (ditto), mock orange, Nootka rose, salal, high bush cranberry, native rhodies, spirea densiflora, evergreen huckleberries.