I noticed that the circled item shows up in the panel whenever there is a maximized window.
I do not like that feature as it changes the position of nearby icons in the panel.
Is there a way to get rid of that "feature" ?
Hi fixit7,
it might be theme related?, what theme are you using and if you have changed themes, does it happen with all of them?. 
I am using the ClearLooks theme.
I will change to another and see what happens.
All the themes have the same issue.
I saw that I can get more themes.
Can I go back to the old one if it does not work?
I have never seen this in many years of using MATE on various platforms. I use Traditional OK, and I use the 'familiar' panel, so maybe that's why.
I don't think it's your theme... what panel layout are you using?
TraditionalOK shows those annoying buttons as well.
Try different panel layouts under MATE Tweak/Panels.
When I went to change layout, it says changing it will completely replace the current..
Are you sure ....lose any customizations et
I think I have done that before and had to spend hours putting programs back in my panels.
You are going to have to change panel layouts if you want to get rid of that behavior. Unless it's an item that can be individually removed - next time you see it, right-click on it and see if there's a 'remove from panel' option - you might get lucky. The notification/indicator system in MATE seems a hot mess to me.
But otherwise, you can save your existing panel layout, try out some others, and maybe you will find one that you prefer. I seriously doubt it will take hours to put your programs back...
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How do I save my existing panel layout?
System/Preferences/Look and Feel/MATE Tweak. The click on 'panels', then 'save as' and give it a name. This way you can always get back to where you were.
Hi @fixit7,
this looks like window buttons applet, which shows your maximized windows' buttons in panel. You shall be able to add and remove it like any other panel applet.
Does right mouse click launch any context menu?
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I never had added it to my panel.
Thanks.
The Redmond theme gets the prize. 
It was preinstalled with Mutiny layout. Maybe some other layout as well.