Sending multimedia content and documents within chat conversations in Microsoft Teams
It is possible to download a video clip in the New Teams Desktop App (at least the way MS Teams is configured in our org).
I came to this forum for the same reason as the op. All I was getting was the same menu options as other posters and could not find a way to download the clip.
However, then I noticed a separate 3-dot menu button located in the upper right corner.
Please note that the method slightly changes once you've started playing the clip.
If you have not started playing the clip, you can simply right-click on the clip and you will see the download option.
If you have already started playing the clip, pause the clip and hover over the clip with your mouse and you should see the 3 horizontal dots (aka meatballs menu or horizontal hamburger menu) icon in the upper right corner of the video clip.
Do not get confused with the other 3-dots menu icon in the lower right. We want the 3-dots icon in the upper right.
Click on the 3-dots button in the upper right corner of the video clip and you should see the "Download video" option.
Hope this helps.