I moved the question to its own thread, so that issue should be resolved 
I’ll start with Why I did it…
Why
Decades ago when I had simple DSL with a small monthly quota, my ISP (iinet) had what they called freezone, or a list of sites/streams (very minimal) which you could visit/watch/listen where bandwidth wasn’t counted, and included their FTP mirror which was an official CD mirror for Ubuntu at the time… That mirror offered Ubuntu Desktop & Server ISOs, so that is what I installed.
I always installed one of those (eg. Ubuntu Desktop), switch the archive site used to also be the ISPs mirror, then would login to text terminal and apt remove ubuntu-desktop and then install the flavor desktop I wanted to try out (eg. apt install xubuntu-desktop) with the package changes being bandwidth quota free due to my using the freezone mirror for downloads. This allowed me explore lots of unusual flavor switches.
I got lazy, and just stopped removing the older desktop, thus my install became multi-desktop, with my selecting which session I’ll use today when I login. (Technically multi-desktop installs wasn’t new for me; I’d been doing it with Debian for years; having first used Debian GNU/Linux before the Ubuntu project even started; you’ve been using Ubuntu I think longer than I have I read earlier this morning, it was part me also exploring the new-to-me Ubuntu)
My current install
I was using a Lubuntu session (LXQt desktop) when I saw your (Xubuntu) Post your desktop post (this was moved from), so I just closed a couple of windows, logged out, selected Xubuntu as my session and entered my password, and I’d switched from using LXQt and a Lubuntu session to a Xubuntu session and Xfce desktop to get my screenshots.
This install was was made with Lubuntu media (though I often do need to check that out)
guiverc@d7050-next:~/snap$ cat /var/log/installer/media-info
Lubuntu 25.10 "Questing Quokka" - Daily amd64 (20250603)
after which I added ubuntu-desktop and tried to add xubuntu-desktop (or maybe didn’t add xubuntu-desktop & discovered a problem then; sorry I forget when this package issue arose on this install & don’t care enough to explore logs to confirm when …) but resolved that by using different metapackages to get Xfce installed, either then or shortly after (whenever issue appeared)
current issue on this install: dracut vs initramfs
Ubuntu Desktop, Lubuntu have switched to dracut yet Xubuntu hasn’t, thus on this box if I
root@d7050-next:~# apt install xubuntu-desktop
The following package was automatically installed and is no longer required:
dracut-core
Use 'sudo apt autoremove' to remove it.
Installing:
xubuntu-desktop
.. snip ..
REMOVING:
dracut
so for now I’ve not got xubuntu-desktop installed here, just other Xfce metapackages; but that issue is newish, and hope that issue will be resolved prior to resolute’s release.
this install; 2025 or older?
Whilst this install may not seem old given the ISO date (20250603) of install media (it was the current daily); that was a non-destructive install and the install actually contains much older data (some back to 2017 or artful cycle when I started using the development release)
(I treat it as a ~mid 2017 install; the box then a 2009 optiplex that was updated to current 2017 optiplex when the 2009 box’s PSU died; data was restored to current box but I don’t believe I restored everything)
This install is dual boot; it contains an ~unused Microsoft Windows 11 (box came with, shrunk to ~50GB) and two Ubuntu installs; this is my primary install which runs whatever release is in development, and my alternate system is a LTS (noble or 24.04). As the development release is unstable, if something breaks I’ll just reboot if I need to & use the LTS release that gets its packages updates etc about each fornight (10-15 days) when I force myself to reboot & switch/update it.
I have other boxes around, this is my newest box and is a 2017 dell optiplex, but my other hardware is much older than this. A second box I use in the evenings or when its hot (its under a ceiling fan) is a 2008 dell optiplex so it’s old Core2Quad with less RAM.
Another note with multi-desktop installs; the 2008 Dell has a Debian forky install (plus 3 Ubuntu releases!) with Debian offering me 16 session choices at login; which is most desktops for me to play with, and less than a year ago I had 26 session choices installed, but I trimmed down and removed the ones I just wasn’t using (most sessions are WMs, and not desktops).
I’ve been unable to bloat Ubuntu as heavily as Debian allows, and tend to stick to 3 flavors usually, or if I really want 4 on an install, I test out order of install sometimes to get the best result; which is what I did on my primary box back in 2017 when I set it up… I have written about it on an askubuntu question, but I didn’t find the answer I wanted; one like it can be read here, and issues encountered will be release specific, but its rare to expect issues on release-upgrade… in fact the dracut issue I’ve touched on in this ‘opus’ reply is the first real package issue I’ve encounted since my actual (as I see it) ~install in 2017.
Your use case
If you wanted to have both Xubuntu (Xfce) & Ubuntu-MATE (MATE) on the same install; I’d expect no problems, as they’ll be pretty easy to manage together too (both are GTK based desktops; where as LXQt/KDE Plasma are Qt and there are some added complexities when GTK & Qt are involved)
If you have questions; ask… otherwise its me rambling on.