This server is a legacy server with light use, hence so many DBs/tables.
In short, what it happened is that the upgrade from the 5.5 MySQL Debian Jessie official package to the 5.6 MySQL Oracle´s version raises the limit of the open files for table caching.
Whilst the open table cache is by default 512 files in 5.5, in 5.6 seems to be at least 2048 files.
Evidently, the problem only manifests itself when you have more than 1000 and so tables already used/opened as the limit per process of files open by non-root users is 1024 by default.
The cache seems to be adjusting itself to around 10% of the open files allowed for the user after raising the limits, will investigate it further.
As final comments, either rising the system limits or adjusting the variable open_files_limit in my.cnf or as a parameter for MySQL solves the problem.
An alternative to raising the system limits in /etc/security/limits.conf, would be then adding in the mysqld section of /etc/mysql/my.cnf
open_files_limit = 100000
It should be duly noted that raising the open file limits is recommended for database, web or email servers that have a lot of tables/vhosts/email users.
The DB responsiveness also seems faster, however I have not conducted any comparative performance tests between the two versions in my particular case.