the region of Bordeaux is a major area of production (mostly for red wines) and is also an appellation (a registered name for wines produced in the area)
Bordeaux wine is produced in Bordeaux from grapes grown there, although grapes from other regions are sometimes illegally used.
They grow the grapes from which Bordeaux wines are produces on vines in vineyards. Wine isn't grown but the grapes are.
E & J Gallo is the winery that makes Thunderbird.