Answer
Does removing metadata reduce image quality?
It does not have to, and here it never does. But a great many tools remove metadata by re-saving the image, and re-saving a JPEG is lossy every single time.
Updated
Metadata and pixels live in different parts of a file. In a JPEG the metadata sits in marker segments before the compressed scan data; in a PNG it is in named chunks alongside the pixel chunks; in a WebP it is a chunk in a RIFF container. Removing it is a matter of dropping those sections and copying the rest through unchanged. The compressed image data is never decoded, so there is no opportunity for it to degrade.
That is not how most tools do it. The easy implementation is to decode the image into a bitmap, throw the metadata away by not carrying it over, and encode a fresh file. This is a few lines of code and it works on every format at once, which is why it is so common. It is also lossy: every JPEG decode-and-resave cycle discards a little more detail, and doing it repeatedly is visibly destructive.
You can usually spot the difference without any tooling. A re-encoded file comes back a noticeably different size from the original in a way that has nothing to do with the metadata you removed, because the whole image was recompressed at whatever quality setting the tool happened to pick.
One case looks alarming and is not. HEIC files are cleaned by overwriting the metadata in place with zeros rather than by deleting bytes, because the format records exact byte offsets for every item and deleting anything would invalidate all of them. The cleaned file therefore comes back the same size as the original. The metadata is genuinely gone; its former space is simply zeroed.
The full write-up covers the surgery for each container, and this comparison measures what re-encoding actually costs.
More on this: Metadata remover, Supported formats, How it works.
Terms used here: Chunk, Lossless, Re-encoding.