As far as I am aware, in Ubuntu MATE the only logging is standard Linux logging.
If you want to prune these logs (and other rubbish that accumulates, although not to the same rate as Windows) bleachbit is your friend.
The privacy settings in “stock” Ubuntu are necessary because it has facilities (certain features of the Dashboard, online accounts with direct access to the OS) which Ubuntu MATE doesn’t have
(Note that the fixubuntu Web site contains a script which, when executed, turns off all the “stock” Ubuntu settings considered as possibly violating privacy; however, it runs into the equally thorny issue of “should you execute a script from a third-party Web site?”).