I'll suggest using apt full-upgrade to ensure all upgrades are applied. There are cases where apt upgrade can leave packages behind.
From man apt you'll see the following
full-upgrade (apt-get(8))
full-upgrade performs the function of upgrade but will remove currently installed packages if this is needed to upgrade the system as a whole.
ie. if packages need to be removed from your system before later packages can be installed; upgrade will leave them uninstalled... allowing you to install them when you decide using apt full-upgrade.
Any 22.04 system with all upgrades applied reported itself as 22.04.1 during the weekend before actual 22.04.1 release; as the .1 upgraded packages had been rolling out for awhile, with the name change of 22.04 to 22.04.1 being the last change.
As usual, this point release includes many updates, and updated installation media has been provided so that fewer updates will need to be downloaded after installation.
ie. the primary purpose is for new installs; so they don't need to install with older media (as you & most people had already done) then applied all upgrades to get there as you did. They could clean install & not have the huge upgrades of packages required with older media.
(FYI: the only reason I use that announcement is my browser knows of it without looking... I'm on the Ubuntu News team & actually posted it... why it's easy for me to grab/use as example)
re: Best Practice... If you apply all upgrades regularly; you were using it days before the release actually occurred !
I ran apt full-upgrade and it seems that it completed successfully. When I put my cursor over the "Menu" button on the task bar on the left side, it shows Ubuntu 20.04.5 LTS. So I'm a bit confused. I thought this was an upgrade to 20.04.01. Also, when I am notified that I have a security update through "Software Updater", I am still prompted to "Upgrade". I canceled the upgrade but it showed that there were 1875 MB to download. I believe the upgrade didn't happen. Can you tell me what I'm missing?
apt full-upgrade upgrades your system fully, but does not change you to another later release.
Ubuntu uses different tools to perform that task, which is covered in different notices on the upgrade process such as here.
Your initial post says you were on 22.04 (in title) which is how I responded, instead of suggesting do-release-upgrade & other release-upgrade tools.
For Ubuntu-MATE 20.04 upgrade instructions (most links I've used till now are generic Ubuntu posts; as I'm a Ubuntu News team member those are easiest for me) can be found here in the section titled
You can upgrade to Ubuntu MATE 22.04 LTS from Ubuntu MATE either 20.04 LTS or 21.10. Ensure that you have all updates installed for your current version of Ubuntu MATE before you upgrade. ...