if you want Ubuntu Mate, sudo apt install ubuntu-mate-desktop, just the mate desktop can be installed by sudo apt install mate-desktop-environment, an easy way to tell if it is MATE is to right click on the panel and select about panel it should say MATE panel, if you no longer want XFCE it can be removed, but the best way to get the full Ubuntu MATE experience is to do a clean install
Yes now I see that I do have mate. The reason I was confused was that I had downloaded the iso for Mint XFCE for my ancient mini laptop (because xfce is lightweight), and it has both xfce and MATE on the same download page:
But now I see that for my desktop PC I also got the official MATE it from the ubuntu mate site here. It’s a bit confusing that there are two downloads for MATE on two sites…
MATE on its own is just a desktop environment. Other distributions like Debian and Solus offer MATE variants of their distributions too.
The only thing in common with Linux Mint (MATE) and Ubuntu MATE is that they use Ubuntu as its base, but both distributions have different objectives and communities.
Heck, you could install XFCE on top of Ubuntu MATE if you like, but for cleanliness, it would be advised to install Xubuntu from scratch.