Answer
Does removing metadata from a video re-encode it?
It does not have to, and here it does not. The metadata boxes are removed and the audio and video streams are copied through bit for bit.
Updated
MP4 and MOV files are containers: a tree of boxes, some describing the file and some holding the encoded audio and video. Metadata occupies its own boxes. Removing it means dropping those boxes and correcting the sizes and offsets that reference them — the encoded streams themselves are copied across without being decoded.
This matters more for video than for images. Re-encoding a clip is slow, and it is visibly destructive in a way that compounds: every generation loses detail, and motion and fine texture suffer first. A tool that "cleans" a video by transcoding it is charging you real quality for a metadata edit.
Offset correction is the part that separates a working implementation from a broken one. Boxes in these containers reference each other by byte position, so removing bytes from the middle invalidates every offset after that point. Rewriting the index rather than merely deleting the box is what keeps the file playable.
The practical check is the same one that works for images: compare durations and stream properties before and after. A file that has only had metadata removed plays identically and differs in size by roughly the size of what was taken out. A re-encoded file differs by a great deal more.
The video metadata remover does this in your browser, so large clips never get uploaded anywhere.
More on this: Video metadata remover, How it works.
Terms used here: Re-encoding.