I'm not sure if you are asking about the micro four thirds standard or the 4:3 aspect ratio, so I'll answer both:

• Why do mirrorless cameras use micro four thirds?

They don't, only Panasonic and Olympus build micro four thirds cameras, there are a lot of other companies making mirrorless cameras (for example: Sony, Fujifilm, Samsung, etc.) that have other mounts.

• Why do mirrorless cameras use 4:3 aspect ratio?

Thomas Edison's lab chose this aspect ratio for silent film, really, nobody knows why but probably because it makes a pleasing image, it's almost square and it's easy to work with.

The SLR/film 3:2 ratio does not come from the golden ratio, when the first roll film camera was made the only roll film available was movie film (4:3) they put the film sideways and used two movie frames for each still frame - two 4:3 frames on top of each other is 4:6 = 2:3

The "standard" aspect ratio closest to the golden ratio was 5:8, it was very unpopular because those proportions simply don't look good (yes, the golden ratio is not automatically the best ratio for everything, who would have guessed)

For everything about aspect ratios read the first answer to this question