If the world had four spatial dimensions, then area would be a tensor?

Roughly speaking, Yes!

In 3 spatial dimensions a 2D thing (an area) uses 2 of the 3 available dimensions. So even though it isn't really a vector (an area should have twice the units of a vector) an area can be described by picking out the single direction that is orthogonal to it.

In 4D an area has a 2D space orthogonal to it, so one needs a 2-index thing (tensor) to specify its orthogonal space, or you could equally use 2-indicies to specify its actual (own) space.

For more details this wiki-page is very clear: https://en.wikipedia.org/wiki/Exterior_algebra

If you have some number of dimensions then a thing (eg. A volume) can be denoted either using either the number of dimensions it has, or the number it is missing. Not only is this why a 2D area in 3D can be specified by a 1D vector (3-2=1), it is also why a Volume in a 3D space can be represented by a scalar (3-3=0). In 4D 3-volume is a vector quantity.