It’s called a highball because of the glass, which is called a highball.
The added height is to allow room for the mixer: water, ginger, or in the case of a white russian – the milk and the ice.

The confusion is that over time, the drink consisting of a jigger of bourbon with ginger on a glassfull of ice, has also been called a "highball." And today, it can refer to a host of other local drinks with the same name. The commonality is that they are all served in the highball glass.

Neat, straight up, and on the rocks are served in lowball glasses, either rocks glass or old fashion.

unless it’s brandy. Then a snifter.

Now, don’t confuse a highball with a collins.
With highball glasses, the inside of the glass should be angled (like a rock glass)
A collins glass is completely cylindrical, thinner than a highball, slightly taller and the glass is usually a bit more delicate.
Also, a highball can be footed, but a collins is never footed. But they’re very similiar glasses. And most places don’t distinguish between the two.

To make it more confusing, highball drinks (that is drinks that are usually served in highball glasses) can be served in double old fashion glasses, which are lowball type glasses, but with more volume than a regular old fashion glass.