This worked for me and was a different path to download a shared video clip in Team chat than normal.
Usually, a clip appears in chat in its own box with a file name and the "three dots" options menu is included, which gives you the option to download (image reference attached).

Sometimes, however, the clip just appears as a playable video within Teams directly, rather than as a file or attachment. Not sure what causes the difference, but when this viewable video clip format appears instead, the only way I found to actually download it (since the menu is not present, nor the download option), is to follow Steven's advice, which was as follows (on an Android phone, at least):
- Use the mobile app version
- Open the Teams chat
- Press and hold the image of the video
- A menu pops up from the bottom with two options: Share Video or Message Options
- Choose the Share Video option and it will access your phone's shareable routes. For me, I have Google Drive set up on my phone, so I was able to "share" the video directly to it, which, in essence, downloaded the video as a file.
This is a ridiculous workaround, but it does indeed work.
I'm not sure why some videos appear as a file and can be played or downloaded, while others appear as a playable video only, with limited options, but here we are.