The terms are medieval:
latitude (n.) late 14c., "breadth," from Old French latitude (13c.) and directly from Latin latitudo "breadth, width, extent, size," from latus "wide," ... . Geographical sense also is from late 14c., literally "breadth" of a map of the known world...
In the fourteenth century, most known-world maps made in the Christian and Muslim West were T & O maps conventionally oriented to place Eden (believed to lie in the extreme East) at the top:
(From Wikimedia.) The black body in the lower middle represents the Mediterranean Sea, with Europe (North) to the left, Africa (South) to the right, and Asia (East) at the top. Breadth (latitude) on this map is north-south and length (longitude) is east-west.
For at least 2300 years it has been well known in the West that the Earth has a globular shape, so in the minds of medieval cartographers and their readers there would have been no confusion between the (flat) shape of the map and the actual shape of the Earth. We deduce these terms are derived from the cartographic point of view of a map reader rather than from descriptive geometry, even though today their definitions are geometric.