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

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.