How to replace Unity by MATE in Ubuntu Xenial 16.04?

Try this:

sudo apt-get install --no-install-recommends ubuntu-mate-core
sudo apt-get install --no-install-recommends ubuntu-mate-desktop

That should ensure you have all the right software installed. Then log into the MATE session and use MATE Tweak to switch away from, and then back to, the “Ubuntu MATE” interface. Then Enable Indicators from MATE Tweak. How that look now?