Thas has not been possible with Postgres json storage type. Instead, the entire JSON blob must be read out, modified, and inserted back in.
This reality is well known to those that understand Postgres, which is why they have HStore. HStore is limited though (particularly to the size of the store), so there is work underway to make it more competitive with MongoDB.
So now they are also releasing a jsonb (b for binary) storage format, which looks promising, but I can't find any information on exactly what its features are. I would love to actually see a benchmark comparing field updates, but this benchmark is not it.
MongoDB is a database with trade-offs, downsides, and more crappy edge cases then MySQL, but it does exist because at its core it allows data modeling that traditional SQL databases are lacking.
MongoDB has first class arrays rather than forcing you to do joins. It supports schema-less data, which is rarely useful, but when you need it can be very useful. It can do inserts and count increments very quickly (yes the write lock means you eventually have to put collections in separate databases), which is also useful for certain use cases.