An open doorway to a different area of the battle was what the spell looked like in previous incarnations in all artwork associated. Some versions showed the doorway as a shimmering portal and some showed it as an open doorway to that showed the location the caster was teleporting to.
Although, as Miniman's answer stated, the only requirements in 5e is a verbal component and thus you don't have to 'step through' the doorway, it is still completely feasible and possible to describe this as a doorway.
As an additional note of inspiration for the name's modern interpretation, in 4th ed's case, teleportation spells took you through other dimensions in order to teleport. The Feywild was a common route, though the Shadowfell was sometimes used. This was written in to the descriptions of several racial abilities. IIRC, Dimension Door included the same description. So in 4e the emphasis was on the dimensional travel and not the door. 2nd edition and prior also related teleportation spells to extra-dimensional travel (though primarily through the Astral Plane).
After looking further I did find one reference, albeit possibly circumstantial, that alludes to the ties between teleportation and dimension travel. The details of the spell Hallow have a specific subheader about 'affected creatures' that states the following:
Extradimensional Interference. Affected Creatures can't move or travel using teleportation or by extradimensional or interplanar means.
This is not specifically stating that teleports are dimensional travel, but the name of Dimension Door and the text of this passage in this spell allude to it highly, as does the fact that spells that discuss warding against teleportation ward against extraplanar travel and vice versa (examples are Forbiddance, Forcecage, and Planar Binding).
While it isn't specifically stated in 5e, the name Dimension Door alludes to traveling through another dimension in order to move quickly, as does the history of the spell and references of other spells. The aesthetics of the spell are primarily up to the player and the DM collaborating, but visual references do typically use the shape of a doorway to represent the spell.