From these articles on Wikipedia:
It seems that after the Meiji Restoration when power was shifted from Kyoto to Edo/Tokyo, the new government started making policy changes in education and the military to unify national speech.
The article mentions a desire to avoid confusion in the military as men were being drafted from various regions with differing dialects as one motivation.
Since the government was seated in Tokyo, it makes sense that the "standard dialect" promoted by the national government would be similar to the native dialect of that area.
Post-WW2 there has been no "official" dialect defined or designated by the government.
However, speech similar to what is common around Tokyo has become tacitly recognized as "standard" by native speakers accross the country largely due to the inherited influence of Meiji-era, Tokyo-centric culture and the development of mass-media like newspapers, television and radio news outlets and publishers of educational material many of which have their own standards.
The 文部科学省 has guidelines for some of this, but it's all related to writing as far as I've seen. There are audio materials used in education, which I'm sure some group of people is approving or rejecting, but I'm not very clear on that.
A good example of this might be NHK's 高校講座 (High School Courses). These are government-supported, publicly broadcast programs that can be used in High School education. I would assume that the speech used in these programs is moderated by the same people who moderate the rest of NHK's broadcasts, but who those people are influenced by is a good question as well.