Is the feature shown in this video actually possible, or is this being done with editing?

https://youtu.be/04nR8jv5VBc?t=261

You could do that with JavaScript in the Browser like here

In the beginning, the code was so slow, that drawing the red line took more than 10 seconds - unintentionally. You can do that also intentionally, of course even slower.

Take a look at GPX Animator. It can take any number of online maps as a background and many of those are OSM based. It is a Java based app that runs on your desktop.

I am not aware of an app which does such an animation. Likely done in video/image editing software.

Possible or editing in what? Eg ArcGIS Pro has animations. Haven’t tried animating in QGIS yet.