Our clients usually have a lot already running on InnoDB so we always have similar setups (preferably identical except the same volume of data) to what they have on the side. Can't tell you how many problems we've caught that way and how much it simplifies things during deployment. If starting anew, it's Postgres.

Our internal stuff is almost all running straight Postgres but a small part of our application stack is abstracted on top of it to provide document DB functionality.

Mysql cannot do parallel query. That's a single query won't spin off multiple threads and work in parallel. This makes it a bad solution for say aggregating over large data. Mysql can handle lots of threads(queries) in parallel making it an ideal solution for web.

Postgres can do parallel query.