Through some our customer engagements at Kasten we see folks who are actually going with both of the models described above - (1) using lots of small databases embedded in each application or microservice (more common with folks building out new microservice style applications from scratch) or (2) Using Kubernetes to actually build out larger centralized databased. You can find a more detailed discussion of this at https://kasten.io/databases/. @Justin_Hartman happy to share some more detailed observations if that will be helpful as you are looking to move some of your environment. Have seen quite a bit of Mongo and Postgres in this space.
Agree with @jberkus that as storage options and support for stateful workloads continue to evolve we will see a lot more databases on Kubernetes as people look to further consolidate their infrastructure.
I also expect to see more solutions (operator based or unifying framework focused https://github.com/kanisterio/kanister) that aim to simplify the ongoing management of such workloads, whether we are talking about many small instances or fewer larger instances.