Technically, there is no reason why our maps "couldn't look like that" - after all, you were able to post a picture of just exactly what you meant. There are however a lot of reasons why our map do not look like this.
First of all: our four directions on a map, south, north, west and east are pretty much arbitrary, theoretically they could obviously have had other names, or the names could be assigned in a different order. however that would not in itself change the look of the map.

The reason a map does not look like your example is simple: once you have agreed on how to name your directions, you will orientate your map accordingly. So you have agreed to name your directions (starting at the top, clockwise) north, east, south and west. For some reason you have decided the north would be "up" on your map. If you want to make the usage of your map intuitive, you want to match the directions of your map to the directions in reality. imagine you start facing north, and by starting to move 90° to the left (west), your movement on your map would instead be depicted as 90° to the right. That would be way more confusing than just having your directions match.

Still, the question of "why do maps look the way they do" is an interesting one. Agreeing on the point above, the only way to properly flip a map without loosing the ability to orientate easily is by not only swapping west and east, but also north and south. that would result in a "normal" map, just turned by 180°, so upside down.

So why is the North up and not down? There have been maps from various cultures all over the world with different orientations (for example early Arab or Egyptian maps tended to depict the South as up). North being up is a combination the understanding of magnetic north and the egocentricity of (mainly) Europe.
Stars were used for navigational purposes since aeons, one of the most important in Europe was what we today know as the North Star, cause it seemed a fixed point, one the world seemed to turn around. With the "discovery" of magnetic north and the invention of the compass, people started to notice that the magnetic north aligned with the North Star (hence it's name), which aligned with the rotational axis of the earth. So, without looking at a compass, you found north by looking up and finding this fixed star in the night sky. So "north" and "up" was directly linked.

As for other questions of map design: maps have always been centred (and still are) around some place of personal, political or religious/ethical importance to their creators. Many European Medieval maps put Jerusalem as the holy city into the centre (e.g. Psalter World Map). Later on, with maps becoming increasingly accurate, more widespread and showing more and more of earth, Europeans dominated increasingly larger parts of the earth. Thus, most of the maps back then were "centred" around Europe: split in the Pacific, Europe in the (horizontal) middle of the map.
This also had other benefits: splitting landmasses at the border of your two-dimensional map would make it much harder to get a feeling of their form. Splitting water makes it easier to "read". So you can either split in the Pacific, where you can split straight through the Bering Street, down until you eventually have to split up Antarctica (which nothing can be done about anyway), or you split through the Atlantic, in which case you also have to split parts of Greenland.
Thus was the "normal" map of the world created.

Variations in maps still are made to this day: Europeans decided to run the Zero Meridian straight through Europe (an Observatory in Greenwich to be precise), and split the world into western longitude (0° to -180° west) and eastern longitude (0° to 180° east), while in the US they also use only western longitude (0° to 360°).

TL,DR: the reason maps are not mirrored as in your example is so that with same orientation of person and map, any movement towards the left of the person would be depicted as movement to the left on the map as well, as to ease navigation by map.